Relevance of star-disc encounters
in massive stellar clusters
From gas-embedded to dissolving populations
INAUGURAL-DISSERTATION
zur
Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fakultät
der Universität zu Köln
vorgelegt von
Manuel Steinhausen
aus Solingen
Max-Planck-Institut für Radioastronomie Bonn
2013
Berichterstatter/in:
Prof. Dr. Susanne Pfalzner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prof. Dr. Joachim Krug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tag der mündlichen Prüfung:
14. Mai 2013
Zusammenfassung
Die aktuelle Vorstellung von Sternenstehung besagt, dass Sterne nicht einzeln sondern
bevorzugt in Sternhaufen entstehen. Als Folge des Entstehungsprozesses sind die jungen Sterne anfänglich von zirkumstellaren Scheiben umgeben, welche aus Staub und Gas
bestehen. Bisherige Untersuchungen des Einflusses von Vorbeiflügen anderer Sternhaufenmitglieder auf diese Scheiben konzentrierten sich ausschließlich auf die frühe Phase der
Sternhaufenentwicklung (< 3 Millionen Jahre), während der die Sterne noch im Gas aus
der Entstehungsphase eingebettet sind. In dieser Arbeit wurde der Einfluss eines störenden
Sterns auf das Stern-Scheibe System während der gesamten ersten 15 Millionen Jahre der
Entwicklung untersucht, also nicht nur die frühe, eingebettete Phase sondern auch die
Expansionsphase des Sternhaufens nach dem Ausstoß des umgebenden Restgases.
Für die eingebettete Phase wurden speziell die Auswirkungen unterschiedlicher Massenverteilungen innerhalb der Scheiben auf Verluste durch Sternvorbeiflüge untersucht. Obwohl der Einfluss der Massenverteilung auf die Scheibenverluste in Einzelstößen sich teilweise stark unterscheidet, ist die Anzahl der Sterne mit gestörten Scheiben im Sternhaufen
weitesgehend unabhängig vom anfänglichen Dichteprofil. Die Ausnahme bilden sehr dichte
Sternhaufen, bei denen Scheibenmasseverluste in den Kernregionen von 40% für steile
Masseverteilungen denen von 60% für anfänglich flache Masseverteilungen gegenüber stehen. Der Grund liegt in der Dominanz der massearmen Sterne in diesen Sternhaufen,
welche aufgrund ihrer üblicherweise massiveren Stoßpartner die größte Abhängigkeit von
der anfänglichen Massenverteilung zeigen.
Nachdem das Restgas aus dem Sternhaufen ausgestoßen wurde, verringert sich dessen
stellare Dichte schnell. Dabei dehnt sich die dichte Kernregion um einen Faktor zehn aus
und die Sterne des dünn bevölkerten, äußeren Randes verlassen den Sternhaufen. Somit
nimmt die Zahl der Stöße deutlich ab und im Allgemeinen werden nur noch sehr wenige
Scheiben während dieser Phase zerstört.
Eine Folge dieser dynamischen Entwicklung ist, dass die große Menge der stellaren Stöße
in der Kernregion von eingebetteten Sternhaufen deutlich die beobachtete Scheibenhäufigkeit
der im Sternhaufen verbleibenden Sterne prägt und somit eine nicht vorhandene Abnahme
der Scheibenhäufigkeit mit dem Sternhaufenalter suggeriert. Der große Anteil der Sterne,
die den Sternhaufen verlassen und in die Feldpopulation übergehen, wird wenig von den
umliegenden Sternen beeinflusst, wodurch Planetenbildung in diesen Scheiben wesentlich
begünstigt ist.
Abstract
Observations reveal that most stars do not form in isolation but as part of a star cluster.
Initially, the young stars constituting these clusters are surrounded by circumstellar discs.
Previous investigations concentrated on the consequences of stellar interactions for these
circumstellar discs during the early phases (< 3 Myr), where the cluster is still embedded
in its natal gas. By contrast, the relevance of star-disc encounters during the entire first
15 Myr of massive cluster development has been investigated in this work, including the
gas-embedded and the cluster expansion phase.
In the embedded phase, the focus was on the influence of the initial shape of the disc-mass
distribution. Although it has a significant impact on the relative disc-mass and angular
momentum losses in certain single star-disc encounters, the fraction of stars with perturbed
discs turns out to be fairly unaffected by the initial density profile. The exception are dense
cluster environments, where disc destruction rates in the crowded core regions are 40% and
60% for steep and shallow disc-mass distributions, respectively. Here, the interactions
of low-mass stars dominate, which show the largest dependency on the initial disc-mass
distribution due to the generally high encounter mass ratios.
After the expulsion of the residual gas the stellar density drops rapidly so that the number
of encounters is considerably lower, and very few discs are completely destroyed. The dense
cluster core region expands by a factor of ten while most of the stars in the sparse cluster
outskirts become unbound. A consequence of this cluster expansion is that the multitude
of stellar encounters in the core regions of embedded clusters significantly shapes the disc
properties of the remnant bound population, whereas the stars joining the field population
are to a much lesser degree affected by encounters.
The expansion process strongly influences the observed disc fractions since it mimics a
non-existent decrease with cluster age. Stars that are dispersed in the field most likely
maintain their discs for a substantially prolonged time span and are, thus, more suitable
for forming planetary systems.
Contents
1 Introduction
1
1.1
Isolated star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Structure of circumstellar discs . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Dissolution of the circumstellar discs . . . . . . . . . . . . . . . . . . . . . .
6
1.4
Interactions with the stellar environment . . . . . . . . . . . . . . . . . . . .
10
1.4.1
The Orion Nebula Cluster . . . . . . . . . . . . . . . . . . . . . . . .
12
1.4.2
Dynamical cluster evolution . . . . . . . . . . . . . . . . . . . . . . .
13
1.4.3
Mass segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Dominant cluster mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.5.1
Embedded cluster phase . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.5.2
Gas expulsion phase . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.5
2 Methods
2.1
2.2
29
Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.1.1
30
Cluster dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modification of nbody6
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.2.1
Identification of encounter events . . . . . . . . . . . . . . . . . . . .
37
2.2.2
Instantaneous gas expulsion . . . . . . . . . . . . . . . . . . . . . . .
39
2.3
Cluster setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.4
Diagnostic aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3 Disc-mass distribution in star-disc encounters
49
3.1
Setup and method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.2.1
Surface density distribution . . . . . . . . . . . . . . . . . . . . . . .
53
3.2.2
Relative mass loss . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.2.3
Relative angular momentum loss . . . . . . . . . . . . . . . . . . . .
58
3.2.4
Adapting a fit formula dependent on the initial disc-mass distribution 60
3.2.5
Definition of disc-perturbing encounter . . . . . . . . . . . . . . . . .
3.3
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
63
3.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Embedded clusters in virial equilibrium
4.1 Setup . . . . . . . . . . . . . . . . . . . .
4.1.1 Density scaling . . . . . . . . . . .
4.2 General cluster dynamics . . . . . . . . .
4.3 Encounter dynamics . . . . . . . . . . . .
4.4 Effect of the initial disc-mass distribution
4.4.1 Dependence on stellar density . . .
4.4.2 Dependence on stellar mass . . . .
4.5 Discussion . . . . . . . . . . . . . . . . . .
4.6 Summary . . . . . . . . . . . . . . . . . .
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5 Cluster expansion phase
5.1 Setup and method . . . . . . . . . . . . . . . . .
5.2 Cluster survivability . . . . . . . . . . . . . . . .
5.3 General cluster dynamics . . . . . . . . . . . . .
5.4 Importance of encounters . . . . . . . . . . . . .
5.4.1 Dependence on stellar mass . . . . . . . .
5.4.2 Evidence for disc destruction . . . . . . .
5.4.3 Dependence on distance to cluster center
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . .
5.6 Summary . . . . . . . . . . . . . . . . . . . . . .
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6 Conclusion
107
7 Summary
111
A Analytical investigations of star formation efficiencies
113
B Numerics of individual star-disc encounters
117
C Stellar number density profiles
123
D Gas expulsion models
127
Bibliography
131
1 Introduction
In accordance with currently accepted star formation scenarios, observations show that
most, if not all, stars are initially surrounded by a circumstellar disc. There is increasing
observational evidence that with time these discs become depleted of gas and dust and
eventually disappear. A variety of physical mechanisms have been found to contribute to
this evolutionary process, one of them gravitational star-disc interactions. In contrast to
previous investigations of such stellar encounters (Clarke & Pringle, 1993; Hall et al., 1996;
Boffin et al., 1998; Pfalzner, 2004; Olczak et al., 2006) this study involves the treatment of
arbitrary disc-mass distributions for the circumstellar discs. Due to the implementation of
a flexible numerical scheme this is realized without the need for explicit simulations of the
entire parameter space.
The importance of stellar encounters is induced by the fact that most young stars do
not form in isolation but as part of a cluster environment. In general only a minor fraction
of the Giant Molecular Cloud Complexes is converted into stars, which results in a protostellar cluster population that is initially surrounded by its natal gas. The residual gas is
expelled by stellar winds, UV radiation and early supernova explosions of the massive stars
within remarkably short timescales. Due to the gas expulsion the binding energy of the
cluster is significantly reduced and the cluster starts to expand. On average, 90% of the
clusters are completely dispersed into the field after about 10 Myr, only a small fraction
remains bound as a stellar cluster (Lada & Lada, 2003). Even though this expansion plays
an important role in the cluster evolution, the effect of stellar encounters under the predetermined conditions of an expanding cluster environment has not been treated so far. In
the second part of the thesis, the importance of stellar encounters in such varying cluster
environments will be investigated for different initial disc-mass distributions.
After about 10 Myr an average of 90% of clusters are completely dispersed into the field
while only a small fraction remains bound as a stellar cluster (Lada & Lada, 2003). However,
the effect of stellar encounters under the predetermined conditions of an expanding cluster
environment has not been treated so far. The focus in the second part of the thesis is on
the gravitational interactions of stellar members in such varying cluster environments.
Moreover, an evolutionary sequence for the density-radius relation of the most massive
clusters in the Solar neighbourhood (Mcluster > 103 M⊙ ) has been recently established by
1 Introduction
Pfalzner (2009). The gas-embedded stellar populations expand rapidly during and after
the gas is expelled ending up as sparse (leaky) OB associations after > 20 Myr. Here, for
the first time the influence of such expanding cluster environments on stellar encounters
will be presented. Various gas expulsion models have been configured that account for the
different evolutionary phases of the stellar populations.
The aim of this work is to obtain an insight in the relevance of stellar encounters in
authentic cluster environments.
1.1 Isolated star formation
The formation process of a single star is relatively well understood. The generally accepted
picture begins with the collapse of a cold molecular cloud and ends when the star has
stopped accreting its circumstellar material, entering the so-called zero-age main sequence
(ZAMS). This period is very short compared to the entire lifetime of a star, given by
4
lifetime ≈ 10
mstar
M⊙
−2.5
Myr.
(1.1)
where the mstar is the stellar mass given in M⊙ .
The youth of stars was suggested by their concentration near associations of high-mass
stars. These have to be necessarily young (Ambartsumian, 1954), because due to their high
masses and consequently high gravitational pressure the hydrogen burning will only last
for a few Myrs. So the presence of high-mass stars demonstrates that star formation is an
ongoing process in our Galaxy.
In Figure 1.1 a sketch of the star formation process including timescales and size relations
is shown. Observations have found that giant molecular clouds (GMCs) with masses of
102 − 106 M⊙ , temperatures of 10 − 20 K and sizes of 10 − 100 pc (Larson, 2003) are
potential sites of star formation. They are concentrated in the spiral arms of galaxies,
surrounded by less dense envelopes of atomic gas. Molecular clouds with a mean density of
≈ 100 H2 molecules per cm3 (Blitz, 1993), entail atomic hydrogen to preferentially associate
into molecular hydrogen H2 (∼ 90%), but there are also some other molecular structures
found like CO2 . The abundance of molecules is strongly dependent on their formation rate
on the surface of dust grains and thus increases with density. The formation of a star from
a molecular cloud requires the resisting forces of turbulent motion, thermal gas pressure
and magnetic fields to be smaller than the gravitational forces. Thus in particular the
cold and dense molecular cloud regions are potential seeds for the star formation processes
(Fig. 1.1a).
Such molecular clouds are typically of very low density, which means that they have to
contract by a factor of several magnitudes to form a star. This enormous reduction in
2
1.1 Isolated star formation
size means that any small initial rotation of the star-forming cloud results in large angular
velocities due to angular momentum conservation, eventually causing the formation of a
planetary disc. In summary, molecular clouds are transient structures that form, evolve and
disperse very quickly, all in a time comparable to ∼ 10 Myr (Hartmann, 2003; Elmegreen,
2000).
Figure 1.1: The process of star formation. Taken from Greene (2001).
The gravitational instabilities of the molecular cloud are a prerequisite for the formation
of so-called protostars. The mass necessary to obtain such gravitationally unstable density
fluctuations is the Jeans mass:
MJ = 5.57c
3
r
ρ
G3
(1.2)
with the isothermal sound speed c and uniform density ρ. The Jeans instability leads to the
formation of a core that starts to accrete material from the surrounding envelope (Fig. 1.1b).
The accompanying release of kinetic energy heats the medium so that temperature and
pressure at the center of the protostar increase. As its temperature approaches thousands
of degrees, the protostar becomes visible as an infrared source.
During the initial collapse, the clump is transparent to its own radiation and the collapse
proceeds rather quickly. With increasing density the clump becomes opaque, restraining
the energy loss by infrared radiation. As a result, temperature and pressure in its center
will start to ascend. At some point, the pressure stops the infall of gas onto the core and
3
1 Introduction
the object becomes temporarily stable as a protostar (Fig. 1.1c). The protostar has initially
about 1% of the final stellar mass and continues growing by accreting material from the
envelope via its protostellar disc.
One consequence of the collapse is that these evolved young stars, in case of m1 ≤ 2 M⊙
known as T Tauri stars, are usually surrounded by massive, opaque circumstellar discs
(Fig. 1.1d). Although the envelope disappears quickly - for a solar mass star the protostellar
phase lasts about 100 000 yrs - disc material can still be accreted onto the stellar surface.
As a by-product of the star formation process planets might form from the debris circumstellar disc. Such pre-main sequence (PMS, Fig. 1.1e) stars radiate energy both from
the heated disc mainly in the infrared and from the accretion of material onto the stellar
surface at optical and ultraviolet wavelengths. A fraction of the material accreted onto the
star is ejected perpendicular to the disk plane in a highly collimated stellar jet. A PMS
star can lose up to 50% of its mass before reaching the main sequence.
This occurs typically after a few million years when thermonuclear fusion begins in the
core and a strong stellar wind is produced which stops the infall of new mass. Since
the circumstellar disc eventually dissipates, there must be other mechanisms to remove
angular momentum from the disc. For example the formation of planets (Fig. 1.1f), binary
or multiple systems or interactions with the stellar environment (Larson, 2009) are possible
explanations.
1.2 Structure of circumstellar discs
There is increasing observational evidence that most, if not all, stars are initially surrounded
by a circumstellar disc consisting of gas and dust. In the past decades such discs have been
detected from optical, infrared and millimeter photometry of young stars. Measurements of
long-wavelength bands showed an excess of emission above that expected from the stellar
photosphere. This excess was interpreted as radiation emitted by a disc-like structure,
which is heated via reprocessing of the central star’s light and accretion luminosity.
The most efficient way to observe stellar discs is examining the L-band excess (Haisch
et al., 2001b), which refers to the near infrared. For example, Lada et al. (2000) found with
this method a fraction of 80 − 85% of all stars in the Trapezium cluster to be surrounded
by discs. Here, the observed wavelength regime is of major importance since it restricts
the observations to a certain part (temperature) within the disc.
Figure 1.2 illustrates the wavelength regions for the spectral energy distribution (SED) of
a flaring protoplanetary disc. The model indicates a near-infrared bump due to the inner
rim of the disc, infrared dust features from the warm surface layer, and the underlying
continuum emission from the deeper (cooler) disc regions. Typically the near- and mid-
4
1.2 Structure of circumstellar discs
Energetic domain
Disk
Star
Wien domain
Rayleigh−Jeans
domain
Figure 1.2: Build-up of the SED of a flaring protoplanetary disc (rdisc = 100 AU). Scattering is not included here
(Dullemond et al., 2007).
infrared emission originate from small radii, while the far-infrared comes from the outer
disc regions. The (sub-)millimeter radiation is mostly emitted from the mid-plane of the
outer disc. Most observations are performed with the Spitzer Space Telescope and limited
to < 10 AU from the central star while a few discs in the Orion Nebula Cluster have
been resolved beyond 50 AU in the optical with the Hubble Space Telescope. Recently,
the ground-based Atacama Large Millimeter Array (ALMA) program started, which will
focus on millimeter and sub-millimeter wavelengths and, thus, the more distant regions
of circumstellar discs. In fact, a much lower disc fraction is found on scales of > 50 AU
compared to observations within distances of less than a few AU to the central star (e.g.
Clarke, 2007).
Disc-mass distribution
To date, observational limitations have prevented a determination of the typical surface
density of protoplanetary discs and its temporal development. This means a unique predetermined initial state for the disc-mass distribution is not well known. A wide variety of
surface densities have been derived by fitting resolved millimeter continuum or line emission
data with parametric disc structure models (e.g. Mundy et al., 1996; Lay et al., 1997) or
in combination with broadband spectral energy distributions (SEDs) (Wilner et al., 2000;
5
1 Introduction
Testi et al., 2001; Akeson et al., 2002; Kitamura et al., 2002; Andrews & Williams, 2007b).
While those studies have profoundly shaped our knowledge of disc structures, all have
fundamentally been limited by the low angular resolution of available data.
The standard fitting method is based on the assumption that the surface density Σ has
a simple power-law dependence of the form
Σ(r) ∝ r−p ,
(1.3)
out to some cut-off radius (e.g. Andrews & Williams, 2007a). Estimates based on numerical
power-law models fitted to observational data lead to distribution indices p ranging roughly
from p = 0 to p = 2. Recent studies have found even unexpected results, e.g. Isella et al.
(2009) who measured distribution indices of p < 0.
Many theoretical studies use a crude approximation of the primordial solar disc (the
Minimum Mass Solar Nebula, or MMSN) as a reference point, which is constructed by
augmenting the current planet masses to match solar abundances, and then smearing those
masses into concentric annuli. The result is fitted with a radial power law, Σ ∝ r−1.5
(Weidenschilling, 1977). Recently, Chiang & Laughlin (2012) determined a similar result
for the Minimum Mass Extrasolar Nebula (MMEN) constructed from the Kepler catalog
of extrasolar planets. They find a slightly steeper surface-density distribution, Σ ∝ r−1.6 .
Analytical approaches also propose different disc-mass distribution indices. The most
widely used model is that of a steady-state viscous accretion disc with a surface density
distribution index of p = 1 (e.g. Hartmann et al., 1998). However, simulations of the
evolution of protostellar discs that form self-consistently from the collapse of a molecular
cloud core yield a surface distribution index of p = 1.5 (Lin & Pringle, 1990; Hueso &
Guillot, 2005; Vorobyov & Basu, 2007), while studies that include magnetised disc material
have found a flatter disc-mass distribution of p = 0.75 (Shu et al., 2007).
Despite the large range of existing distribution parameters upcoming observational results from ALMA will potentially provide further insights on the disc structures.
1.3 Dissolution of the circumstellar discs
With time, the protoplanetary discs surrounding new born stars become depleted of gas
and dust and eventually disappear (Haisch et al., 2001b; Hillenbrand, 2002; Sicilia-Aguilar
et al., 2006; Hernández et al., 2007; Currie et al., 2008; Hernández et al., 2008; Mamajek,
2009; Massi et al., 2010). The remnant circumstellar material will either be accreted onto its
central star, dispersed into the interstellar medium, or aggregated to large protoplanetary
bodies. In Figure 1.3 the disc fraction in stellar clusters, which are conglomerations of
thousands of stars, are shown as a function of the mean cluster age. The cluster disc
6
1.3 Dissolution of the circumstellar discs
fraction (CDF) is a fundamental quantity, which is defined by the number of disc-less stars
divided by the total number of stars and is often used as a global tracer of a clusters
dynamical state. In general, observations indicate a strong decrease of the cluster disc
fraction with increasing cluster age. Most of the discs are completely dissolved after about
6 Myr.
Figure 1.3: Cluster disc fraction as a function of the cluster age. The solid line shows an exponential fit. The Figure
and data references are taken from Mamajek (2009).
The cluster disc fraction is not only a function of the cluster age but depends likewise on
the stellar mass of the disc-surrounded star, the binary fraction, and the cluster density.
Uniform observations of these effects are rather challenging, due to the large distances of
even the nearest star forming regions (> 100 pc Wilking et al., 2008) and drastic observational limitations. In most cases the clusters are still embedded in their natal gas and,
therefore, observations are restricted to long wavelength selections and sub-samples of the
brighter high-mass stars, which are preferentially located in the central cluster regions.
Despite these limitations, it was found that disc lifetimes decrease for higher stellar
masses (Hillenbrand et al., 1992; Lada et al., 2000; Stolte et al., 2004; Carpenter et al.,
2006; Lada et al., 2006; Kennedy & Kenyon, 2009) with dispersion timescales of only a few
105 yr for pre-main sequence stars of 2 − 8 M⊙ - so-called Herbig Ae/Be stars (Alonso-Albi
et al., 2009). Kennedy & Kenyon (2009) analysed stellar disc fractions in nine young clusters
by observing the infrared excess, a tracer for circumstellar discs heated by the central star,
7
Disc fraction for stellar mass range
1.5 - 7.0 Msun
1 Introduction
Disc fraction for stellar mass range
0.6 - 1.5 Msun
Figure 1.4: Shown are the disc fractions according to accretion signatures (×) and infrared excess (+) for two
stellar mass bins. The dashed line indicates an equal disc fraction for the mass bins. The data results
from observations of (1) Taurus, (3) IC348, (4) Tr37, (5) NGC2362, (6) OB1bc, (7) UpperSco, and (9)
NGC7160 are presented. The Figure has been adapted from Kennedy & Kenyon (2009).
and the equivalent widths of Hα emission, which is a signature for accretion. Figure 1.4
shows their results by comparing the disc fraction of sun-like stars (0.6 − 1.5 M⊙ ) to the
disc fraction of stars > 1.5 M⊙ , using an equivalent number of stars for each mass bin. It
can be seen that circumstellar discs around sun-like stars seem to survive longer than discs
around high-mass stars. For lower masses they find no significant deviations. However, in
other cases the disc fraction seems to decline towards very low stellar masses (Lada et al.,
2004, 2006; Oliveira et al., 2006), which is mostly interpreted as sun-like stars providing
the most favorable conditions for planet formation. Note that due to the large error ranges
for disc fractions of low-mass stars these results have to be treated with caution.
Another significant influence on the cluster disc fraction is provided by binary or multiple
stellar systems. Bouwman et al. (2006) found much shorter lifetimes of circumstellar discs
around stars that are part of a binary system than those for single stars. According to
their investigation the mean disc dissipation time for binaries is found to be about 5 Myr
while discs around single stars survive roughly 9 Myr. Furthermore, Cieza et al. (2009)
distinguished between tight and loose binary systems. They observed twice as much discs
around binary systems with separations of 40 − 400 AU compared to the number of discs
around hard binaries with separations < 40 AU.
Furthermore, observations reveal that the cluster disc fraction depends strongly on the
8
1.3 Dissolution of the circumstellar discs
cluster density. Luhman et al. (2008) compare the disc fraction of the sparse cluster
Chamaeleon I to the one of the denser stellar population of IC 348, both with cluster
ages of about 2.5 Myr. For masses > 1 M⊙ they found significantly less dissolved discs
in Chamaeleon I. Hence, they concluded a decreasing disc lifetime for high-density star
forming regions.
100
Starburst cluster
Sparse associations
Haisch et al. (2001)
NGC 2024
Trapezium
cluster disc fraction
80
CrA
MBM12
60
Taurus
E CHA
IC 348
40
NGC 2264
NGC 6618
R 136 / 30 Dor
η CHA
σ Ori
20
TW Hya
NGC 3603
NGC 2362
Arches
0
0
2
4
6
8
10
12
14
time[Myr]
Figure 1.5: Shown is the cluster disc fraction as a function of the cluster age in Myr. The values for the starburst
clusters (red) are taken from Stolte et al. (2010) (and references therein) while the sparse associations
(blue) are taken from Fang et al. (2013) (and references therein). The remaining data points are taken
from the sequence found by Haisch et al. (2001b) (and references therein). In most of the previous
cases minor or even no errors have been specified for the cluster ages and disc fractions. Taking into
account additional observations by several authors (Casey et al., 1998; Webb et al., 1999; Weintraub
et al., 2000; Sung & Bessell, 2004; Nisini et al., 2005; Meyer & Wilking, 2009; Currie & Sicilia-Aguilar,
2011; Kudryavtseva et al., 2012) the error bar estimates have been improved and now reflect an increased
parameter range.
This would also suggest a spatial dependence of observational results. Stolte et al.
(2010) surveyed the cluster disc fraction of B-type stars in the Arches cluster, one of the
densest stellar populations in the Milky Way (ρcore > 105 M⊙ pc−3 ) containing > 100 Otype stars (Figer et al., 2002). They found a strong dependence of the disc fraction on
the radial distance from the dense cluster center with an increasing excess fraction for
larger radii. Similar results have been obtained for the starburst cluster NGC 3603 where
the disc fraction increases with radius from 20% (r < 0.6 pc) to 40% (0.8 < r < 1 pc)
(Stolte et al., 2004). Similar results have been obtained for stellar populations in the Solar
neighbourhood, like the Orion Nebula Cluster (Hillenbrand et al., 1998).
9
1 Introduction
In Figure 1.5 a combination of the established cluster disc fraction from Haisch et al.
(2001b), Stolte et al. (2010) for massive compact starburst clusters, and the recently published results from Fang et al. (2013) representing sparse associations are shown as a function of the cluster age. Error estimates have been considerably improved by seeking for
additional observational age and disc fraction data in the literature (see caption of Fig. 1.5).
Apart from the generally expected decline of the cluster disc fraction with time, a significant difference in the slope emerges for the plotted density groups. The red data points
represent the scope of the starburst clusters in which the circumstellar discs disperse very
quickly (< 4 Myr). Stolte et al. (2010) interpreted these findings as a result of the high
masses of the stars in their sample. Note, that the disc fraction of the inner core of 30
Doradus has not been resolved, which implies that the shown disc fraction might be an
upper limit. While the data by Haisch et al. (2001b) is dominated by low-mass stars, starburst cluster observations are mostly limited to OB-type stars, due to their high densities
and large distances. Fang et al. (2013) presented a study of sparse stellar associations
covering various stellar mass ranges. As shown in Figure 1.5 by the blue crosses the cluster
disc fractions of these stellar populations decrease much slower than the previous. After 6
Myr it is unclear if the discs will completely disappear or if a fraction of around 20% will
maintain. Regardless of the stellar mass a dependence of the cluster disc fraction on the
cluster density can be identified. While in dense clusters the discs disappear within a few
Myr, circumstellar discs in sparse associations might survive for at least 10 Myr.
1.4 Interactions with the stellar environment
It is currently unclear which physical mechanism dominates the evolutionary disc destruction processes. Among the great variety of effects are internal processes such as viscous
torques (e.g. Shu et al., 1987), turbulent effects (Klahr & Bodenheimer, 2003), and magnetic
fields (Balbus & Hawley, 2002), but as well external disc destruction processes like photoevaporation (Scally & Clarke, 2001; Clarke et al., 2001; Matsuyama et al., 2003; Johnstone
et al., 2004; Alexander et al., 2005, 2006; Ercolano et al., 2008; Drake et al., 2009; Gorti
& Hollenbach, 2009) and tidal interactions (Heller, 1993; Clarke & Pringle, 1993; Ostriker,
1994; Heller, 1995; Hall et al., 1996; Hall, 1997; Larwood, 1997; Boffin et al., 1998; Pfalzner,
2004; Pfalzner et al., 2005a; Moeckel & Bally, 2006; Kley et al., 2008).
Since it is long known that stars form not only in isolation but in stellar populations,
like the Orion Nebula Cluster (ONC), star cluster regions became of major importance
for studying disc destruction via external processes. Here, investigations seem to indicate
that photoevaporation should by far dominate the disc destruction (Scally & Clarke, 2001;
Balog et al., 2008). During this process the perturbation of protoplanetary discs results
10
1.4 Interactions with the stellar environment
from the UV radiation from massive stars. Hereby, the radiation interacts with the disc
matter and thus accelerates preferentially light disc elements outwards. Another significant
destruction mechanism, although often discarded as probably unimportant, is the influence
on circumstellar discs by stellar encounters, which has already been suggested in the 90s
by e.g. Clarke & Pringle (1993); Hall et al. (1996); Boffin et al. (1998). At that time first
numerical simulations of selected stellar fly-by scenarios were performed, which already
yielded substantial effects on the shape of the circumstellar discs.
The focus of the present numerical investigation will be on the effect of such gravitational
star-disc interactions on the disc-mass distribution and, therefore, the mass and angular
momentum of the discs. A star-disc encounter∗ can cause matter to become unbound,
be captured by the perturbing star or pushed inwards and potentially be accreted by the
central star. The extent to which this happens depends on the periastron distance, the
mass ratio of the stars, the eccentricity and, moreover, the initial (pre-encounter) mass
distribution in the disc.
Star-disc encounters with focus on the mass and angular momentum losses due to gravitational interactions in ONC-like cluster models have been reviewed in many investigations
prior to this study (Pfalzner, 2004; Olczak et al., 2006; Pfalzner & Olczak, 2007). In particular, Olczak et al. (2006) found that star-disc interactions influencing the circumstellar
discs are more frequent in the inner core of the ONC than previously assumed (Scally &
Clarke, 2001). Moreover, the massive stars act as gravitational focii which results in a rapid
destruction of their discs due to multiple perturbations by lighter stars (Olczak et al., 2006;
Pfalzner et al., 2006).
However, most previous numerical studies of star-disc encounters have used only a single
density distribution, mainly focusing on the case of a theoretically motivated r−1 disc-mass
distribution (Hall et al., 1996; Hall, 1997; Pfalzner, 2004; Olczak et al., 2006; Moeckel &
Bally, 2006; Pfalzner & Olczak, 2007). Star-disc encounters with different initial disc-mass
distributions have only been considered in a very limited way. Heller (1995) performed
numerical simulations of two different mass distributions (p = 0 and p = 1.5) concentrating
on parabolic encounters with equal mass stars, while Hall (1997) investigated initial surface
distributions of p = 0 and p = 1 for close and penetrating encounters of unity mass ratio.
A study of a wide parameter range focusing on multiple initial disc-mass distributions still
needs to be performed.
Nevertheless, numerical studies of star-disc encounters only allow a sectional view of
the processes since it is impossible to simulate each combination of encounter parameters.
Earlier analytical studies by Ostriker (1994) did not suffer from this shortcoming. In
∗
Here, the term ’star-disc encounter’ describes encounters in which only one of the stars is surrounded
by a disc, in contrast to disc-disc encounters, in which both stars are surrounded by discs.
11
1 Introduction
her study, a first order approximation of the angular momentum loss dependent on the
initial disc-mass distribution is given. However, the validity of her results is limited to
large periastron radii (for example rperi /rdisc > 3 for M2 /M1 = 1), where the angular
momentum loss is usually well below 10 %. Close or even penetrating encounters cannot
be interpreted by this linear perturbation theory (Ostriker, 1994; Pfalzner et al., 2005b)
making numerical studies indispensable in this regime.
Taking the huge variety of observed and theoretical motivated disc-mass distributions
into account (Section 1.3), one has to consider different initial disc-mass distributions to
evaluate their effect in star-disc encounters. In this work, the effects of star-disc encounters
are investigated for a large parameter space considering most configurations that can be
expected in a typical young cluster. The investigated mass distributions cover the entire
range of the so far observed disc-mass distributions.
1.4.1 The Orion Nebula Cluster
Visible • WFPC2
Infrared • NICMOS
Trapezium Cluster • Orion Nebula
WFPC2 • Hubble Space Telescope • NICMOS
NASA and K. Luhman (Harvard-Smithsonian Center for Astrophysics) • STScI-PRC00-19
Figure 1.6: Two views of the Trapezium cluster in the Orion Nebula from the Hubble Space Telescope. The image on
the left, an optical spectrum image taken with Hubble’s WFPC2 camera, shows a few stars shrouded in
glowing gas and dust. On the right, an image taken with Hubble’s NICMOS infrared camera penetrates
the haze to reveal a swarm of stars as well as brown dwarfs. The figure is taken from Luhman & O’Dell
(2000).
The Orion Nebula Cluster (ONC) has been chosen as one model cluster for our simulations since it is the nearest massive young star cluster and one of the best observed young
dense star forming regions. In Figure 1.6 it can be seen that large parts of the cluster are
12
1.4 Interactions with the stellar environment
covered by the remnant gas, which prevents extensive observations in the visible regime.
However, in the infrared band the star-forming region, which consists of about 4000 stars
with masses above the hydrogen-burning limit of 0.08 M⊙ in a volume of 2.5 pc (Hillenbrand & Hartmann, 1998; Hillenbrand & Carpenter, 2000) and half-mass radius rhm ≈ 1 pc,
can be resolved. It is located south of Orion’s Belt, along our spiral arm of the Milky Way,
at a distance of 414 ± 7 pc to our Sun (Sandstrom et al., 2007; Menten et al., 2007) and
is part of the Orion Molecular Cloud Complex. The mean age of the cluster has been
estimated to be about 1 − 2 Myr (Hillenbrand & Hartmann, 1998; Palla & Stahler, 1999).
Most of the stars have already been formed and the residual gas has been ejected to the
outer parts.
Besides most of its objects being T Tauri stars, there is also strong evidence for the
occurrence of protostars embedded in a dusty envelope, which indicates that the star formation process is still ongoing. The inner region of the ONC emerges as a very bright
and dense part of the cluster, called the Trapezium, which already expelled most of its
gas. It is this cluster region where most of the massive stars are located. The most massive (msystem ≈ 50 M⊙ ) and also brightest (2.5 · 105 L⊙ ) stellar system in this region is the
binary θ1 C Ori, which is one out of four leading high-mass objects that ionize the whole
nebula by their intense ultraviolet radiation. The strong radiation of these stars also shapes
the Orion proplyds, protoplanetary discs that are surrounded by tear-drop shaped ionization fronts (O’dell et al., 1993). Finally, the high densities in these inner cluster region
(ρ0.053 pc = 4.7 · 104 pc−3 , McCaughrean & Stauffer, 1994) suggest that stellar encounters
might be relevant for the evolution of circumstellar discs within the Trapezium region.
1.4.2 Dynamical cluster evolution
The dynamical evolution of the stellar population is largely determined by gravitational
interactions of the stellar members. Hereby, a fundamental quantity describing the global
dynamical state of the cluster is the virial ratio
Q=
T
rvir σ 2
=
,
|W |
GMtot
(1.4)
where T is the total kinetic energy, W the total potential energy of the cluster, σ is the
statistical dispersion of the velocities (so-called velocity dispersion), G is the gravitational
constant, Mtot is the total cluster mass, and rvir is the virial radius. For an equal-mass
system rvir is defined by the average distance between the particles:
1
rvir
=
1
|ri − rj |
.
(1.5)
i6=j
13
1 Introduction
In general, the magnitude of the virial ratio specifies whether a cluster is contracting
(Q < 0.5), expanding (Q > 0.5) or in virial equilibrium (Q = 0.5). This classification can
be inferred from the virial theorem, which yields for any stable spherical system consisting
of N self-gravitating particles
2T + W = 0.
(1.6)
While individual deflections of the stellar trajectories are unpredictable in such chaotic
N-body systems the overall dynamics of the stellar cluster can be specified by a set of time
scales that are introduced in the following.
Crossing time
The crossing time, tcross , is given by (Heggie & Hut, 2003; Binney & Tremaine, 2008)
tcross =
√
r1.5
2rvir
= 2 2 √ vir
σ
GMtot
(1.7)
p
where σ ≈ GMtot /2rvir and the virial radius rvir is given by the potential energy W as
rvir = GMtot /2|W |. The crossing time estimates the typical time scale a star needs to cross
the cluster and, thus, describes the amount of mixing of the stellar population. In general,
a non-equilibrium dynamical system adjusts back to equilibrium within a few tcross . For
massive clusters consisting of 30 000 stars and a radius of R = 6 pc typical crossing times
of tcross ≈ 0.3 Myr are obtained. If the cluster age is less than tcross the population is close
to its initial state.
Relaxation time
The path of a star is perturbed by several weak deflections of neighbouring stars. A time
that quantifies after which time span these deflections become significant and the star
forgets about its initial path is given by the two-body relaxation time (Spitzer, 1987)
N
tcross ,
(1.8)
ln N
where N is the number of stars. It expresses the time after which the changes of the
direction of motion of the stars become comparable to the initial velocity dispersion. Consequently, within trelax the dynamical cluster state changes significantly and evolves to a
globally relaxed system. Note that for the usual case of N > 35 we obtain trelax > tcross so
that the cluster might be mixed and close to dynamical equilibrium but is not relaxed. In
other words, for high particle numbers the cluster dynamics are less influenced by encounters due to the deeper cluster potential.
trelax = 0.1
14
1.4 Interactions with the stellar environment
The relaxation time within a cluster varies significantly between the innermost and outermost parts. Therefore, it is useful to define a half-mass relaxation time (Spitzer, 1987;
Binney & Tremaine, 2008)
trh
0.14N
=
ln(γN )
s
3
rhm
,
GMtot
(1.9)
where rhm is the half-mass radius, containing the innermost half of the total cluster mass,
and γN the argument of the Coulomb logarithm (Spitzer & Hart, 1971) with γ = 0.11
for equal-mass particles (Giersz & Heggie, 1994) and < 0.11 otherwise (Giersz & Heggie,
1996). While the dynamics of old globular clusters with typical ages of tage ≈ 1010 yr and
relaxation time scales of trelax ≈ 108 yr are significantly influenced by stellar interactions,
young clusters like the ONC with relaxation time scales trelax = 1.5 · 107 yr > tage =
106 Myr (adapted from Hillenbrand & Hartmann, 1998) can be approximated as collisionless systems with their dynamics being determined by a smoothed gravitational potential.
However, although the global dynamics are rather unaffected by stellar interactions the
circumstellar discs properties can be significantly altered by stellar fly-bys.
Ejection and evaporation time
Stars are able to escape from a bound stellar population if their total binding energy
Etot = W + T ≥ 0. This translates into the minimum stellar velocity needed to escape the
cluster potential, so-called escape velocity, given by
vesc (r, t) =
p
2|φ(r, t)|.
(1.10)
The potential φ(r, t) splits into a part given by the total enclosed mass within r, Mtot (r, t),
and a potential contributed by the surrounding material
φ(r, t) = −G
Mtot (r, t)
+ 4π
r
Z
∞
′
′
ρ(r , t)r dr
r
′
.
(1.11)
The second term is obtained by integrating over each radial mass shell at r′ > r.
There are two different types of stellar collisions that force a star to leave the cluster:
(i) Ejections, which require a strong, single encounter event increasing the velocity of one
of the stars to an amount larger than the escape velocity and (ii) Evaporation caused by
several weak interactions that gradually increase the kinetic energy of a star to finally
become unbound from the cluster.
The ejection time scale of the cluster can be approximated by tejec = −N dt/dN ≈ 104 trh
(Henon, 1969) while numerical simulations (Spitzer, 1987) suggest an evaporation time
scale of the order of tevap ≈ 330trh .
15
1 Introduction
Collision time
The collision time, tcoll , defines the time scale for an actual physical contact between two
stars, with collision radius rcoll = 2rstar . The collision rate, 1/tcoll , is obtained by a simple
integration of the number of encounters per unit time over all stellar velocities, which
results in (Binney & Tremaine, 2008)
1
tcoll
−1
√
GM1
2
= 16 πnσrcoll 1 + 2
2σ rcoll
(1.12)
where n is the stellar density, M1 the mass of the central star, and σ the velocity dispersion.
Here, the second term represents the effect of gravitational focusing, which leads to a
reduction of the collision time. For simplicity all stars are assumed to have the same single
mass, which strongly underestimates the effect of gravitational focusing.
For typical quantities like a mean stellar mass of < M1 >= 0.5, velocity dispersion of
σ = 1km/s, rcoll = 2R⊙ , and central densities of n = 5 × 103 pc−3 as found for the ONC
we obtain an collision time scale of the order 1011 yr. Hence, stellar collisions are rather
unlikely for stellar densities investigated in this study.
However, for a gravitational interaction between the circumstellar disc and the encountering star much larger radii rcoll ≫ 2R⊙ can be assumed. Gutermuth et al. (2005) estimated
in their study the collision time scales in dependence of the cluster density. Assuming
two collision radii, rcoll = 103 AU for protostellar envelopes (Motte & André, 2001) and
rcoll = 102 AU for classical T Tauri discs (McCaughrean & O’dell, 1996), they find typical
encounter time scales of the order 105 yr and 107 yr, respectively, for stellar densities of
104 stars per pc−3 .
1.4.3 Mass segregation
Observations showed that stellar clusters, sometimes only a few Myr old, are typically mass
segregated, e.g. the most massive stars of the only 1 Myr old ONC are located within the
inner rcore = 0.3 pc of the cluster (Hillenbrand & Hartmann, 1998). Not only dynamical
evolved clusters like the starburst cluster Arches (Harfst et al., 2010) are found to be
mass segregated but also still embedded clusters show this trend like the ONC, NGC 6231
(Raboud & Mermilliod, 1998) or NGC 1893 (Sharma et al., 2007). There is a strong debate
if the observed mass segregation can be a product of dynamical two-body relaxation of the
cluster or if only primordial mass segregation is capable to explain segregation processes
on such short timescales. A better understanding of the dynamical mechanisms is needed
to further constrain these concepts.
In case of primordial mass segregation massive stars form preferentially at the cluster
center. This process is explained by an increased amount of accretion material as well
16
1.5 Dominant cluster mode
as an accelerated accretion of the surrounding gas due to the location of the stars at
the bottom of the potential well. Bonnell & Davies (1998) investigate the evolutionary
effect of dynamical mass segregation and conclude that the position of massive stars in
young clusters generally reflects the cluster’s initial conditions due to the young cluster
age relative to their relaxation time (see also Hillenbrand & Hartmann (1998)). Therefore,
previous numerical studies assumed an initial mass segregation of the ONC locating the
most massive star in the cluster center and the three next massive stars randomly within
r = 0.6 · rhm (Bonnell & Davies, 1998) so that it fits today’s observations of the ONC.
Another explanation might be a dynamical mass segregation. The wide range of observed
stellar masses (typical 0.08 − 150 M⊙ which will be further detailed in Sec. 2.3) implies that
their initial kinetic energy depends not only on the stellar position in the cluster but also
strongly on the individual stellar masses, with largest kinetic energies for the most massive
stars. Stellar interactions lead to an equipartition of the kinetic energy by massive stars
transferring kinetic energy to the low-mass stars. As a consequence the massive stars sink
towards the cluster center.
Typically, the time scales for dynamical mass segregation are larger than the cluster
relaxation time tequ > trelax . However, recent numerical investigations found that cool
clusters dynamical mass segregate on timescales far shorter than expected (Allison et al.,
2009; Olczak et al., 2011). Also dynamical friction from the molecular gas can effectively
reduce the time scale of the dynamical mass segregation (Er et al., 2009). Moeckel &
Bonnell (2009) show, that for system ages less than a few crossing times star formation
scenarios predicting general primordial mass segregation are even inconsistent with observed segregation levels for some young embedded clusters. Furthermore, Ascenso et al.
(2009) argue that there is currently no observational evidence of mass segregation in young
clusters since there is no robust way to differentiate between true mass segregation and
sample-incompleteness effects.
1.5 Dominant cluster mode
Although the star formation process itself seems comparatively well understood (see Sec. 1.1)
there is currently a serious discussion about the environment within which the majority
of stars form. A main problem is the absence of a general definition of a stellar cluster.
Moreover, the question arises whether most stars form in the few massive clusters, the large
number of small-N clusters or in hierarchically structured environments with arbitrary local surface densities. In the latter case massive clusters would only represent an exception.
The two main formation models will be briefly explained in the following.
17
1 Introduction
Two sequences of clustered star formation
It is generally believed that most stars, if not all, do not form in isolation but as part of a
group of stars consisting between just a few to up to ten-thousands of stars (e.g. Lada &
Lada, 2003; Porras et al., 2003; Fall et al., 2005; Evans et al., 2009). However, today only
a low fraction of stars in the solar neighbourhood is found in clusters. A comparison of the
number of young clusters (< 10 Myr) to the number of evolved clusters indicates that the
majority of clusters dissolves quickly after their formation (e.g. Fall et al., 2005; Pfalzner,
2009). Lada & Lada (2003) determine that 90% of the observed embedded clusters within
2 kpc of the Sun have to disperse within 10 Myr after their formation and about 96% after
100 Myr. They called this rapid cluster disruption and the accompanied lack of evolved
open clusters infant mortality.
The dissolution of young stellar clusters is explained by the expulsion of their residual
gas caused by stellar winds, UV radiation, and the first supernova explosions. Due to
a significant reduction of the total cluster mass a high number of stars might become
unbound. This process was studied by several authors (Hills, 1980; Lada et al., 1984;
Adams, 2000; Geyer & Burkert, 2001; Boily & Kroupa, 2003a,b; Goodwin & Bastian, 2006;
Bastian & Goodwin, 2006; Baumgardt & Kroupa, 2007; Parmentier & Pfalzner, 2012;
Pfalzner & Kaczmarek, 2013) and will be detailed in Section 1.5.2.
1e+06
-3
Cluster density [Msun pc ]
Arches
-3
-3
Westerlund 1
RSGC1
Westerlund 2
DBS2003
RSGC2
Quintuplet
h Per
3
5000 Msun pc /R
> 10 Myr
4Myr < age < 10Myr
< 4Myr
Galactic center cluster
NGC 3606
Trumpler
10000
4
15000 Msun pc /R
STAR BURST CLUSTERS
chi Per
100
Cyg OB2
NGC 6611
IC 1805
NGC 7380
NGC 2244
Ori Ib
1
LEAKY CLUSTERS
Ori Ic
U Sco
Lower Cen Crux
Upper Cen Lup
Ori Ia
I Lac
0.01
1
10
Cluster radius [pc]
Figure 1.7: Cluster density as a function of the cluster size for clusters more massive than 103 M⊙ . The Figure is
taken from Pfalzner (2009) while the observational data points are taken from Wolff et al. (2007), Figer
(2008), Borissova et al. (2008), and references therein.
18
1.5 Dominant cluster mode
While Lada & Lada (2003) concentrated on the temporal evolution of embedded clusters,
recent investigations by Pfalzner (2009) found evidence for a bimodal evolution process of
stellar clusters more massive than 103 M⊙ in the radius-density plane. Figure 1.7 shows
the sequential radius-density-age relation revealing two distinct groups of clusters. Clusters
in the first sequence are termed starburst clusters and consist of several ten-thousand
stars in a rather small volume of only a few 0.1 pc half-mass radius. Within 20 Myr of
development their total masses of ≈ 20 000 M⊙ remain fairly constant while their sizes
increase significantly to about 1 − 3 pc. Pfalzner & Kaczmarek (2013) determined star
formation efficiencies of up to 70% as an explanation for the minor dissolution of the
cluster and comparatively low velocity dispersions of the stellar members (Rochau et al.,
2010; Bastian, 2011; Hénault-Brunet et al., 2012).
Stars within the dense environments of such starburst clusters are very prone to stellar
encounters. Olczak et al. (2012) investigated stellar encounters in the Arches starburst
cluster and determined that more than 30% of all stellar discs are completely destroyed
by gravitational interactions within the first 2.5 Myr. However, well-observed starburst
clusters in the Milky Way are very rare because they are mostly located in the dense spiral
arms or close to the Galactic Center where observations are limited by the high background
density. In the present study the focus is on massive clusters in the solar neighbourhood
and starburst clusters will not be considered.
Massive nearby star forming regions, which are subject of the present study, evolve along
the second sequence in the density-radius plane, the so-called leaky cluster sequence.
These clusters have initial total masses similar to the starburst clusters but are distributed
in a much larger area with a radius of about 6 pc. Within 20 Myrs the size of leaky clusters
increases significantly up to ≈ 20 pc while they additionally lose large amounts of their
total mass. Pfalzner & Kaczmarek (2013) found a star formation efficiency of < 30% of
such clusters and concluded that the expulsion of the residual gas is a major process for
cluster members to become unbound.
Hierarchical cluster formation
In contrast to the observed modes of a clustered formation the size and surface density of a
star forming region in the theoretical context of a hierarchical cluster formation process are
believed to be continuously distributed throughout the hosting galaxies (Elmegreen et al.,
2006; Bastian et al., 2007). In this picture one might argue that the embedded clusters
observed by Lada & Lada (2003) are simply a selection effect and an observational bias
leads to a neglect of the stars residing in the lower local density regimes. Another approach
is that already during the formation process bound stellar systems are naturally formed in
the observed density peaks of the interstellar medium, while lower density clusters instantly
19
1 Introduction
dissolve. Hereby, a minor fraction of ≈ 30 − 35% of all stars is estimated to be born in such
bound entities (Kruijssen, 2012). By contrast Lada & Lada (2003) observed that 90% of
stars are expelled from a cluster after the gas expulsion process and only about 10% might
remain bound.
The probably most popular study about this subject was published by Bressert et al.
(2010) who investigated local surface densities in the solar neighbourhood (< 500 pc distance) and found a smoothed log-normally shape of the stellar distributions. They favour
a continuous star formation process over clustered modes of star formation due to a lack
of multiple peaks in the observed local surface density distribution of star forming regions.
However, recent studies found that their results might be caused by a superposition effect of multiple modes of star formation (Pfalzner et al., 2012; Gieles et al., 2012) so that
a smooth surface density distribution can not be used as a criterion for hierarchical star
formation. Moreover, starburst clusters or the dense inner Trapezium region of the ONC
have been excluded in Bressert et al. (2010) giving no further qualification of the Pfalzner
(2009) cluster sequences.
The discussion if most stars form in clusters leads to the question: What is a cluster?
While Lada & Lada (2003) estimated 90% of all stars to form in clusters due to their cluster
definition of a local surface density threshold of 3 stars per pc2 Bressert et al. (2010) came
up with a density limit of 200 stars per pc2 leading to only 26% of all stars forming in
clusters. The Bressert et al. (2010) limit is based on the estimated density for encounters
to become important in a clustered environment (Gutermuth et al., 2005). However, such
arguments are irrelevant for our study since we focus on the actual cluster densities and
find high variations of the local surface densities between the maximum and the observed
local surface density, which is basically used to define a stellar population as a cluster. A
detailed description will be given in Section 4.3. Additionally, here, the low density regions
observed by Bressert et al. (2010) are of minor importance since stellar interactions might
be rare in their present density conditions.
1.5.1 Embedded cluster phase
For the first 105 to 106 yr of the star formation process the newly formed protostars are
surrounded by their natal gas. Such evolutionary very young clusters are called embedded
clusters. Due to gas extinction the formed young stars are mostly not visible in the optical
regime. Only longer wavelength emissions like near infrared (NIR) radiation are able to pass
obstacles in form of dust and gas providing first insights in T Tauri star characteristics.
Still, little constrains can be made on stellar quantities due to the optical thick cloud
structure.
So far it is unclear if the formation of stars in such embedded clusters starts in the dense
20
1.5 Dominant cluster mode
10000
-3
Cluster density [Msun pc ]
-1.3
~rcl
-3
~rcl
rcl
propagating star formation
< 50 stars
50 - 100 stars
100 - 300 stars
> 300 stars
exposed clusters
-1
100
31.5
M
sun
pc -2(
rc /pc
l
) -1
-4
~rcl
1
0.1
1
10
Cluster radius [pc]
Figure 1.8: Cluster density as a function of cluster radius for embedded clusters with more than 200 observed
members (coloured symbols) and the leaky cluster sequence found by Pfalzner (2009) for clusters more
massive than 103 M⊙ (black open circles). The thin dashed line indicates the observational limit defined
by Carpenter et al. (2000). The original figure is taken from Pfalzner (2011).
inner part of Giant Molecular Clouds or if stars form simultaneously throughout the cloud,
hence, to a much lower degree in the outer regions. Figure 1.8 shows the cluster density
versus the cluster size for the embedded clusters observed by Lada & Lada (2003). Pfalzner
(2011) was the first to interpret their observed mass-radius relation as a sequence, which
contains as well the precursors for the radius-density-age relation of exposed leaky clusters.
If sufficient material is provided by the surrounding molecular cloud the embedded clusters
potentially gain enough mass to form additional stars and thus develop towards the leaky
cluster sequence. Moreover, Pfalzner (2011) shows that the development in the densityradius plane can be fitted by ρ ≈ 100 · r−1.3 M⊙ pc−3 . The density development in the
embedded phase is flatter than the evolutionary track of the leaky clusters since no gas
has been expelled and star formation is still ongoing. After reaching a maximum mass of
several 104 M⊙ and cluster radii of about 1 − 3 pc the residual gas is ejected, the cluster
rapidly expands, and most of its stellar members become unbound (see also Section 5).
While there are predictions that star formation starts simultaneously throughout the
entire Molecular Cloud, the evolution shown in Figure 1.8 suggests that the process of star
formation starts in the central cluster region and is delayed for larger distances from the
center. Figure 1.8 indicates that the first few stars of a forming cluster are located in the
very dense center of the cluster with rcluster < 0.2 pc. However, Parmentier & Pfalzner
21
1 Introduction
(2012) claim that star formation occurs simultaneously within the whole cluster but at a
slower rate in the outer parts. They show that due to an observational threshold, given by
the population of background stars (Carpenter et al., 2000), the distant sparse star forming
regions of the molecular cloud might be neglected in observations (see also Pfalzner, 2011).
Therefor, the results found by Pfalzner (2011) present a density limit for the embedded
cluster evolution in the density-radius plane.
1.5.2 Gas expulsion phase
The rapid disruption of stellar populations within a few Myr after their formation, as
assumed in the concept of infant mortality, is explained by an extremely fast gas expulsion
process. Observations show that the majority of young clusters eject their residual gas in
less than ≈ 5 Myr (Whitworth, 1979; Leisawitz et al., 1989; Lada & Lada, 2003), however,
the significance of the responsible mechanisms is unclear. Adams (2000) estimated the
gravitational binding energy of a cluster to be
46
E ≈ 4 × 10 erg
M
1 000 M⊙
2 R
1 pc
−1
.
(1.13)
Convenient initial quantities, which fulfill the requirements to evolve along the leaky cluster
sequence, would be Mtotal = Mgas + Mstars = 100 000 M⊙ and R = 6 pc (Pfalzner &
Kaczmarek, 2013). According to Equation 1.13 an energy of 1049 to 1050 erg would be
required to expel the entire amount of residual gas. A simple explanation would be an
early supernova explosion of an O-type star. Such energetic explosions release energies
of 1051 to 1053 erg (Kitayama & Yoshida, 2005; Kitaura et al., 2006) within very short
timescales. However, the time interval until a high-mass star collapses in a supernova
explosion decreases exponentially with its mass (Schaller et al., 1992), with typical lifetimes
for massive O-stars (120 M⊙ ) of 3 Myr while the first supernova explosion of θ1 C Ori in
the Orion Nebula Cluster would be expected after about 5 Myr. However, as argued by
Palla & Stahler (2000) rapid gas expulsion occurs even for low-mass clusters in less than
10 Myr, which are void of massive stars.
Other energy sources to explain an early ejection of the gas are given by the UV flux
and stellar winds of the massive stars. Tenorio-Tagle et al. (1986) analysed the effect of
photoionisation in young clusters and found that massive stars are able to remove significant
amounts of gas before the first star explodes in a supernova. Depending on the gas profile
Chiosi & Maeder (1986) estimated an O5-star releasing 4 × 1052 erg in form of radiation
during its lifetime (≈ 10M yr), while stellar winds account for additional 1049 erg until the
star explodes (Kudritzki et al., 1987). See also Goodwin (1997) and Lamers & Cassinelli
(1999) for more details.
22
1.5 Dominant cluster mode
A characteristic timescale unit to evaluate the impact of the cluster gas dispersal is the
crossing time that has been introduced in Section 1.4.2. Baumgardt & Kroupa (2007)
showed that clusters return to dynamical equilibrium already a few tens of crossing times
after the gas is expelled.
Alternative disruption processes
After a rapid gas expulsion process a high fraction of about 90% of all stellar clusters is
dispersed. However, observations indicate a further decreasing cluster frequency over the
first 100 Myr with approximately 90% of clusters being disrupted every decade in age (Fall
et al., 2005; Chandar et al., 2006; Whitmore et al., 2007). Such further disruptions are
caused by mechanisms that become important on much longer time scales than the gas
expulsion.
Relevant internal processes are given by two-body relaxation and stellar evolution, while
tidal fields are potential external effects. For our prototype cluster we obtain a typical
half-mass relaxation time (the time scale a star ’forgets’ about its initial path) of about
100 Myr (Spitzer, 1987), which indicates that due to the high number of stars two-body
relaxation is of minor importance in the early stage of cluster development.
Stellar evolution of individual high-mass stars might reduce the total mass of young
clusters. Depending on its mass the stars evolve on different timescales, where stars more
massive than 8 M⊙ generally resolve quickly in a supernova explosion. However, the first
supernovae explosions of the most massive stars are expected after a few Myr, which is
beyond the scope of interest of this investigation, e.g. it takes > 30 Myr for a 10 M⊙ star to
explode in a supernova (see Eq. 1.1). Goodwin (1997) determined a total cluster mass loss
of 10% within the first 50 Myrs due to stellar evolution. Within 10 Gyr the cluster mass is
reduced by approximately 30% (Baumgardt & Makino, 2003). Hereby, the mass reduction
can be assumed as nearly adiabatically, which minimises the fraction of stars becoming
unbound by a reduction of the potential cluster energy. The effect of stellar evolution on
the time scales considered in this work plays only a minor role and can be neglected.
Moreover, interactions with an external Galactic tidal field become essential when investigating massive starburst clusters like Arches, which is located at a projected distance
of only 25 pc from the center of the Milky Way (Olczak et al., 2012). However, here, the
effect of a tidal field has been neglected since the focus is on the early evolution of massive
embedded clusters in the solar neighbourhood, thus, a sufficient distance to the Galactic
center can be assumed. In such regions the clusters are only marginally affected by weak
tidal fields of the surrounding clouds (see also Sec. 1.5.2).
23
1 Introduction
Star formation efficiency
The star formation efficiency ǫ turns out to be an important criterion for the survivability
of a stellar cluster after the residual gas is expelled. It is defined as ǫ = Mstars /M0 , where
Mstars is the mass of the stellar content while M0 = Mstars + Mgas represents the total
mass of the stars and the surrounding gas cloud. According to its definition ǫ is assumed
constant for a certain Molecular Cloud. In the following the star formation efficiency will
be introduced in detail.
Analytical investigations found that in case of a rapid gas-mass loss of the primordial
embedded cluster star formation efficiencies of > 50% are required to obtain a subsequently
bound system (Hills, 1980; Mathieu, 1983; Elmegreen, 1983). Therefore, they analysed the
evolution of the total cluster energy during the gas expulsion phase. These analytical
models solely consider the total kinetic and the total potential energy of the cluster to
determine the survivability of a bound stellar system. Adams (2000) and Boily & Kroupa
(2003a) reviewed the analytical approach by Hills (1980) and showed that the fraction of
bound stars is not only a function of the star formation efficiency but also of the initial
stellar distribution function. They found lower star formation efficiencies of around 30% to
retain a significant fraction of bound mass. A short introduction of the analytical approach
is given in Appendix A.
Lada et al. (1984) performed the first Nbody simulations of embedded star clusters and
their dissolution after gas expulsion. Their model clusters contained only a minor fraction
of 50 (in a few cases 100) stars per cluster that were distributed in a Plummer shaped
sphere. In their simulations the gas cloud was initially included by a background potential
and removed over varying expulsion timescales of up to 1 Myr. They found that a small
fraction of stars can remain bound as a cluster even for ǫ < 50%, which is in contrast to
the conclusions of Hills (1980). They also determined that the bound fraction depends
strongly on the gas expulsion timescale, with significantly more stars remaining bound for
an increasing expulsion timescale. In general, the expulsion occurs on time scales less than
the clusters dynamical crossing time (Hills, 1980) with a maximum time for the expulsion of
four crossing times (Lada et al., 1984). Several further numerical investigations confirmed
the approach by Lada et al. (1984), whereat some of them are summarized in Figure 1.9.
It is shown that a star formation efficiency of about 30% is the threshold for the survival
of a bound cluster core after rapid gas expulsion (Goodwin, 1997; Lada, 1999; Goodwin &
Bastian, 2006; Baumgardt & Kroupa, 2007). Even lower star formation efficiencies of less
than 10% are possible if assuming adiabatic gas expulsion (Baumgardt & Kroupa, 2007).
Furthermore, Lada et al. (1984) investigated initially sub-viral clusters and found that
the low velocities of the stars can be a potential mechanism to obtain bound clusters in
case of star formation efficiencies being lower than 20%. Geyer & Burkert (2001) continued
24
1.5 Dominant cluster mode
Figure 1.9: The fraction of stars which remain bound as a cluster after gas removal as a function of the star
formation efficiency for results from the literature. While the right line of symbols represents the case
of an instantaneous gas expulsion, open triangles and stars on the left show results for slow gas removal.
The Figure is taken from Baumgardt & Kroupa (2007).
this thoughts by investigating different initial velocity dispersions of a stellar distribution.
They argue that star formation efficiencies have to be high in order to to maintain a bound
core but can be as low as 10% if the initial velocity dispersion is very small.
Geyer & Burkert (2001) used an embedded stellar King distribution of 1000 stars for their
investigations. In particular, they used two different approaches for simulating the initial
gas distribution. Despite using Nbody simulations for the stars and a background potential
modelling the gas, they performed sophisticated SPH (Smoothed Particle Hydrodynamics)
simulations of the gas component. However, similar results were obtained with both models,
favouring the gaseous background potential for most embedded cluster simulations due to
its simplicity.
In most cases numerical investigations focus on standard models of an equal cluster and
gas cloud profile using the star formation efficiency ǫ, the cluster size rcluster , and the gas
expulsion timescale texp as parameters to describe the bound fraction. However, Adams
(2000) published the first investigation assuming a locally varying star formation efficiency
ǫ(r) by simulating the development of a stellar King profile and a nearly isothermal distribution of the gas component. As expected for the collapse of a GMC (see Appendix
in Adams, 2000), the stellar density is higher than the gas density in the cluster center
(ǫ(r) > 50%) whereas the density in the outer regions is dominated by gas (ǫ(r) ≪ 50%).
25
1 Introduction
Adams (2000) found that after the gas is expelled an exceeding fraction of stars remains
bound as a cluster, which can be approximated by Nbound /Ntotal = 2ǫ − ǫ2 .
Moreover, he assumed an anisotropic velocity distribution model and found that the
fraction of stars remaining bound as a cluster is higher than for the isotropic distribution
model. In general, the number of bound stars depends strongly on the velocity distribution function as suggested by Boily & Kroupa (2003a). Chen & Ko (2009) reviewed the
numerical investigations by Adams (2000) and claimed that the initial kinetic energy of the
cluster and the initial cluster-cloud mass ratio at a certain Lagrange radius are much more
sensible criteria for the fraction of bound stars.
Despite the star formation efficiency, stellar distribution function and the gas expulsion
timescale the fraction of bound stars depends on the cluster radius. Baumgardt & Kroupa
(2007) studied the influence of the galactic tidal field by assuming the cluster size to be
much smaller than its distance to the Galactic center. They found that the tidal field
slightly reduces the fraction of bound stars in limited cases of rhm /rtide > 0.05, where rhm
defines the half-mass radius of the cluster and rtide is the radius beyond which the tidal
forces of the hosting galaxy are able to remove stars from the cluster.
Another idea to obtain bound stellar clusters after blowing out the natal gas is the
merging of sub-clusters. Fellhauer & Kroupa (2005) showed that star formation efficiencies
can be as low as 20% and still leave a gravitationally bound stellar population in star cluster
complexes that subsequently merge to a bound population.
Estimates of star formation efficiencies by observations of young stellar clusters are very
challenging. A major difficulty is finding the actual number of young stellar objects, which
are surrounded by dusty discs and deeply embedded in their natal gas. However, observations of Giant Molecular Clouds, the birthplaces of young protostars (see Sec. 1.1),
indicate that star formation appears to be a rather inefficient process. In general, only a
minor fraction of ≤ 30% of the total mass is typically transformed into stars (Lada & Lada,
2003).
Myers et al. (1986) studied 54 Giant Molecular Clouds in the inner galactic region and
found star formation efficiencies ranging from ǫ = 0.06% to 50%. They interpreted the
low median value of ǫ = 2% as a reason for a low fraction of remnant bound clusters
after the expulsion of the residual gas. Additionally, they already concluded that the
considerably varying star formation efficiencies of the clouds depend on the initial cloud
mass. Recent observations by Weidner et al. (2007) confirmed a strong decrease of the
global star formation efficiency from low-mass star clusters to massive clusters containing
high-mass stars.
However, Lada et al. (2010) suggest that the highly non-uniform distribution of stars is
not a function of the initial cloud mass as assumed by Myers et al. (1986) and Weidner
26
1.5 Dominant cluster mode
Figure 1.10: (a) Ratio of the number of Young Stellar Objects, N (YSO), to the total cloud mass as a function of the
total cloud mass. This is roughly equivalent to a measure of twice of the star formation efficiency as a
function of the cloud mass. The figure is taken from Lada et al. (2010). (b) Star formation efficiency as
function of the total cluster mass. The dashed line indicates the luminosity limit. The figure is taken
from Murray (2011) with the plotted data taken from Solomon et al. (1987) (filled triangles), Heyer
et al. (2009) (open triangles), Grabelsky et al. (1988) (filled squares), and Bronfman et al. (1989) (filled
pentagons).
et al. (2007) but varies by more than one order of magnitude (see Fig. 1.10a). Therefore,
they analysed cloud complexes within 450 pc from the Sun and determined star formation
efficiencies of ǫ < 5% (N (YSO)/M ass < 0.1, Fig. 1.10a). Murray (2011) found star
formation efficiencies of ǫ = 8+12
− 8 % in a similar sample of young stellar clusters in the Milky
Way and plotted the efficiency as a function of the total cloud mass (Fig. 1.10b). They
found that the observed trend can potentially be explained by an observational selection
effect, namely a luminosity limit indicated by the dashed line.
Evans et al. (2009) found similar star formation efficiencies of ǫ = 3% to 6% for star
forming regions within 300 pc from the Sun. Considering an average overall cloud lifetime
of about 20 Myr (Matzner, 2002; McKee & Ostriker, 2007; Murray, 2011), they argue that
in case of a continuing constant star formation for 10 Myr a cluster is able to reach final
efficiencies of 15% to 30%. Although most studies focus on clusters in the solar neighbourhood, such little star formation rates have also been determined for extragalactic regions,
where efficiencies of ǫ = 14+18
− 8 % are determined for galaxy cluster cores (McDonald et al.,
2011).
Many studies have investigated the effect of the gas expulsion on the evolution of stellar
clusters. However, here, for the first time the relevance of star-disc encounters during the
27
1 Introduction
entire cluster evolution process will be investigated. The focus will be on the most massive
clusters in the Solar neighbourhood, which - as discussed above - evolve along the leaky
cluster sequence after an early expulsion of their natal gas. The numerics and setup for
the models used in the present investigation will be detailed in the following.
28
2 Methods
2.1 Numerics
The densities of young star clusters span a wide range from sparse clusters like Taurus with
densities of 1 to 10 stars per pc3 (Luhman et al., 2009) to very massive and compact clusters
like the Arches cluster with a core density of several 105 stars per pc3 (Figer et al., 1999).
Such dense stellar environments lead to strong gravitational interactions between the cluster
members and effect as well the protoplanetary discs. To investigate the temporal evolution
of circumstellar discs in such large stellar number environments numerical simulations are
required. Two different types of simulations were used in this context, a parameter study
of individual stellar encounters and dynamical cluster simulations of all (but disc-less)
stars. These numerical investigations can be treated separately and are combined in a final
diagnostic step.
In my diploma thesis I performed a first parameter study to investigate the effect of an encounter between a disc-surrounded and a disc-less star, so-called star-disc encounter events.
Hereby, the stars are represented by two particles and the disc by N pseudo-particles, which
reproduce the dynamical behaviour of the real mass distribution of a circumstellar disc.
The disc particles initially move on Keplerian orbits around the star. The temporal development of the disc-mass distribution during an encounter is modelled using a fifth-order
Runge-Kutta Cash-Karp integrator with an adaptive time-step size control for the numerical calculations. Long-range interactions of the gravitational forces between the disc
particles and the perturbing stars are calculated using a hierarchical tree method (Barnes
& Hut, 1986). The numerical methods are outlined in Appendix B.
Here, the interactions between stars in different model clusters have been simulated for
several Myr of cluster evolution. The code is based on the code nbody6 originally developed by Sverre Aarseth in the late 60s (Aarseth, 1963, 1974, 2003). Apart from a general
presentation of the temporal cluster development, these simulations provide valuable information about typical encounter parameters like encounter frequencies, minimum stellar
separations, stellar mass ratios and orbital eccentricities.
In the following, the code nbody6 will be introduced. Moreover, code modifications and
the combination with the single encounter investigations will be detailed.
2 Methods
2.1.1 Cluster dynamics
In the past, simulations of star clusters containing several thousand particles have been
computationally too expensive but new developments in computer techniques and numerical
algorithms in the last years have lead to a reduction of the computation time and increasing
accuracy, so that computationally intense simulations of massive stellar clusters can be
solved in appropriate time periods today.
Massive stellar clusters contain up to ten thousands of stars that are gravitationally interacting. The dynamics of the system are described by 6N ordinary differential equations,
3N for the particle positions and 3N for the velocities that have to be solved numerically.
The particles interact only via gravitational forces, which are described by a set of N
Newtonian equations of the form:
N
X
mj (~ri − ~rj )
F~i
¨
= −G
,
~ri = ~ai =
mi
|~ri − ~rj |3
j=1
(2.1)
j6=i
where mi and mj are the particle masses, ~ri and ~rj the particle positions, and G is the
gravitational constant. This is known as the gravitational N-body problem, which cannot
be solved analytically for N ≥ 3. For the realisation of appropriate cluster models a highly
accurate N-body code is needed. Here, the code nbody6 developed by Sverre Aarseth is
used, which operates with a high-order integrator for the integration of the gravitational
forces at a high precision.
To improve the computational effort sophisticated numerical schemes like a parallelised
force summation, an improved neighbour treatment and a regularisation for close encounters are implemented. All methods are described in the following (for further details see
Aarseth, 2003).
Hermite integration scheme
The generic problem of solving ordinary differential equations of N-th order can be reduced
to the study of a set of N coupled first-order differential equations for the functions ~ai
(with i = 1, 2, ..., N ) which have the general form
~a˙ i (t) = fi (t, ~a1 , ..., ~aN ), i = 1, ..., N.
(2.2)
The advantage of this scheme is that there are no derivatives on the right hand side of
any of those equations, and there are only first derivatives on the left hand side. In the
following, the numerical treatment of one of the N differential equations will be shown
exemplary.
30
2.1 Numerics
One possibility to approximate a solution within an interval ∆t = t1 − t0 is given by the
Euler method
~ai,1 = ~ai,0 + ∆t · fi (t0 , ~ai,0 ),
(2.3)
where ai,1 is the approximation of the reference point a(t1 ) and the approximation of a(t0 )
is given by ai,0 . For the Euler method the derivative information are only expanded to the
beginning of the next interval, t1 , and therefore the obtained errors are only one power of
∆t smaller. So this linear approximation will lead to large errors and consequently unstable
trajectories.
A more accurate time-integrator to track the trajectories of a high number of stars
for several Myr is the fourth order Hermite integrator scheme which was introduced and
implemented in nbody6 by Makino (1991) (see also Makino & Aarseth, 1992)). It works
with a predictor-corrector method, which means it involves a prediction step for a rough
estimation of the particles path and a corrector step to adjust the initial approximation.
The first step is to further differentiate Equation 2.1 and obtain
~a˙ i,0 = G
N
X
j=1
j6=i
mj
~vji,0
(~rji,0 · ~vji,0 )~rji,0
−3
3
|~rji,0 |
|~rji,0 |5
,
(2.4)
where ~rji,0 = ~rj,0 − ~ri,0 is the current relative position and ~vji,0 = ~vj,0 − ~vi,0 the current
relative velocity between the particles i and j. A first prediction of the new position ri,1
and velocity vi,1 is simply given by their Taylor series:
~a˙ i,0
~ai,0
(t1 − t0 )2 +
(t1 − t0 )3
2
6
~a˙ i,0
= ~vi,0 + ~ai,0 (t1 − t0 ) +
(t1 − t0 )2 ,
2
~ri,1 = ~ri,0 + ~vi,0 (t1 − t0 ) +
~vi,1
(2.5)
(2.6)
where now ~ai,0 = ~v˙ i,0 denotes the current acceleration of particle i.
A Hermite interpolation can be used to further improve the predicted values. Therefor,
the Taylor expansion for ~ai,1 and ~a˙ i,1 are generated to third and second order, respectively:
~ai,1
...
¨i,0
~a i,0
~
a
= ~ai,0 + ~a˙ i,0 (t1 − t0 ) +
(t1 − t0 )2 +
(t1 − t0 )3
2
6
...
˙~ai,1 = ~a˙ i,0 + ~a
¨i,0 (t1 − t0 ) + ~a i,0 (t1 − t0 )2
2
(2.7)
(2.8)
where ~ai,0 and ~a˙ i,0 are known from Eq. 2.1 and Eq. 2.4, respectively. While ~ai,0 and ~a˙ i,0
¨i,0 and
can simply be obtained by the initial position and velocity, a direct calculation of ~a
...
~a i,0 is not possible. However, Eq. 2.7 and Eq. 2.8 can now be combined to obtain the third
order derivative:
31
2 Methods
...
~ai,0 − ~ai,1 ~a˙ i,0 − ~a˙ i,1
+
~a i,0 = 6 2
(t1 − t0 )3
(t1 − t0 )2
!
.
(2.9)
Substituting Eq. 2.9 into Eq. 2.7 gives the second order derivative of the acceleration:
˙
˙
¨i,0 = −6 ~ai,0 − ~ai,1 − 2 2~ai,0 − ~ai,1 .
~a
2
(t1 − t0 )
t1 − t0
(2.10)
These solutions can now be used to estimate the corrections of Eq. 2.5 and Eq. 2.6 by an
expansion to two higher orders:
5
4
...
¨i,0 (t1 − t0 ) + ~a i,0 (t1 − t0 )
~ri = ~ri,1 + ~a
24
120
(2.11)
(t1 − t0 )3 ... (t1 − t0 )4
+ ~a i,0
6
24
(2.12)
¨i,0
~vi = ~vi,1 + ~a
where ~ri and ~vi denote the corrected position and velocity, respectively.
The advantage of the Hermite predictor-corrector scheme is - beside its comparatively
simple structure - an eminent energy and angular momentum conservation, which is required for a long timescale cluster development. This provides the opportunity to determine
the cluster dynamics at high accuracy∗ .
Individual time steps
The stellar densities in the cluster center can be several magnitudes higher than in the
cluster outskirts. Many close interactions in these inner parts require a high spatial and
temporal resolution. In principle this would result in decreasing time steps for all cluster
regions, which would significantly increase the computation time. One vital method to
reduce the computational effort for such systems is to adapt the integration accuracy to
the physical problem. In the temporal domain this is realised by individual time steps ∆ti
for the particles, so that in case of large dynamical timescales the integration steps are
not as frequent as required for close particle interactions. Shorter individual time scales in
the dense cluster regions assure a sufficient accuracy while the computational effort can be
reduced by prolonged time steps in the sparse cluster parts.
The individual time step method is organised as following. First, the particle i of minimum time step width ∆ti is determined. Afterwards the positions of the remaining particles
for this particular time step are predicted (as introduced in the Hermite scheme, Eq. 2.5)
to obtain the gravitational force acting on i. After adopting the position, velocity and time
∗
Here, simulations with deviations in the total energy of > 0.001% have been excluded from the diagnostic step.
32
2.1 Numerics
for particle i the process starts again. In nbody6 the individual time step is defined by
(Aarseth, 1985)
∆ti =
¨i | + |~a˙ i |2
|~ai ||~a
...
η
¨ i |2
|~a˙ i || ~a i | + |~a
!1/2
,
(2.13)
where η is a dimensionless accuracy parameter which is typically chosen to be 0.02 (Aarseth,
2003).
A so-called block time-step scheme is used in nbody6 to obtain shared integration time
steps for the particles. Hereby, the individual time steps are quantised by powers of 2,
∆ti = 2−k
k = 0, 1, 2 . . . n,
(2.14)
where n provides the quantisation exponent for the smallest time step ∆ti,min > 2−n . Each
individual time step, given by Eq. 2.13, is truncated to the closest block time step given
by Eq. 2.14.
particle
4
3
2
1
time step
Figure 2.1: Sketch of a block time step scheme.
Figure 2.1 shows a sketch of the block time step scheme. As outlined above, particle 1
starts with the shortest time step after which its new accurate position and velocity will
be determined. Particle 4 defines the largest step after which the phase space coordinates
of all particles will be quantified and updated. For the last time step bin the particles 1,
2, and 3 have equal time step sizes. Here, the minimum time step is four times larger than
the initial minimum time step.
33
2 Methods
Neighbour treatment
Gravitational N-body simulations are conceptually simple in principle, because they merely
involve integrating the 6N ordinary differential equations defining the particle motions
in Newtonian gravity. However, in practice, the number of particles N involved can be
very large (as in the present application) and the number of gravitational particle-particle
interactions, which have to be evaluated, rises as
N X
i−1
X
i=1 j=1
1=
N (N − 1)
= O(N 2 ) .
2
(2.15)
Similarly to the usage of adaptive time steps for the temporal domain, the accuracy
in the spatial domain can be adopted to the physical problem, reducing the required N 2
calculations. A possible solution is to construct an algorithm which restricts the direct
force summation to particles close to each other, while the contribution from more distant
objects is considered on a less accurate level.
To significantly reduce the number of calculation steps the so-called Ahmad-Cohen neighbour scheme, introduced in detail by Makino & Aarseth (1992), is implemented in the
nbody6 code. Here, the general idea is to divide the total acceleration on a star ai in two
parts, a distant (ai,d ) and a nearest neighbour force contribution (ai,n ):
ai = ai,d + ai,n .
(2.16)
The distant part, ai,d , denominates the slowly varying force of very distant particles while
the nearest neighbour part, ai,n , describes the high fluctuating potential of particles in the
immediate vicinity of particle i. While the regular forces are typically weak the irregular
forces strongly influence the particle.
Figure 2.2: Sketch of a time sequence of the Ahmad-Cohen neighbour scheme for one particle taken from
Makino & Aarseth (1992).
These two forces are now updated after different time intervals ∆tn and ∆td with ∆tn <
∆td . As indicated in Figure 2.2 at certain irregular time steps, tn1 , tn2 , tn1′ , tn2′ , . . . , only
34
2.1 Numerics
the nearest neighbour forces are calculated to higher orders while distant particle forces
are predicted by Eq. 2.5 and Eq. 2.6. At regular time steps t0 , t1 , t2 , . . . all forces are
determined to high orders.
The specification if a particle is a distant or a near neighbour particle is given due to
its distance to the investigated particle rn , which is determined at each regular time step.
Aarseth (1985) defined the adjusted radius rnnew as
rnnew = rnold
n 1/3
p
n
(2.17)
which is defined by n the number of nearest neighbours and np the predicted number of
nearest neighbours, which is given by np = nmax (0.04C)1/2 . Here, C is the density contrast
C ∝ n/rn3 and nmax the maximum number of nearest neighbours for which 2N 1/2 has
proven to be a good estimate in case of large N (Aarseth, 2003).
rn
1/3
2 rn
buffer zone
Figure 2.3: Sketch of the neighbour scheme for one particle. Particles with an irregular force contribution are shown
as filled circles while open circles indicate regular force contributors. The yellow particle in the center
represents particle i.
Moreover, particles within a buffer zone between rn and the radius 21/3 rn , which fulfill
the requirement ~rij ~vij < 0.1rn2 /∆td , are included in the list of nearest neighbours to assure
a consideration of fast approaching particles. The classifications of the nearest neighbours
in the radial sphere are illustrated in Figure 2.3.
Two-body regularization
Although above described methods reduce the computation time significantly, the investigations for close interactions like short period binary systems or very eccentric stellar
35
2 Methods
orbits can considerably increase the computational effort. The small particle separations
also result in strongly increasing gravitational forces, which lead to numerical difficulties.
One way to overcome this obstacle is a regularisation of the close interacting particles. In
nbody6 this is achieved by the so-called KS-regularisation introduced by Kustaanheimo
& Stiefel (1965) and explained in detail in Aarseth (2003).
The binary is treated separately from the neighbour-force summation scheme. External
perturbations are implemented in the binary’s equation of motion by an external tidal field
acceleration ~r¨ext :
~
~¨ = − M R + ~r¨ext ,
R
R3
(2.18)
~ = ~ri − ~rj is their relative
where M = mi + mj is the total mass of the binary system and R
−2
coordinate. To avoid the singularity of type r , a substitution of Eq. 2.18 by a differential
time transformation of the form
dt = Rn dτ
(2.19)
is generated, resulting in:
′′ ~ ′′
~
~ ′′ = nR R − M R + R2n~r¨ext ,
(2.20)
R
R
R3−2n
where here n = 1 is chosen according to Sundman (1912) (for more details see Aarseth,
2003). In the following the external perturbation will be neglected and the one dimensional
regularisation case will be exemplary discussed. These constraints lead to a reduction of
Eq. 2.20:
x′′ = x′2 /x − M.
(2.21)
The binding energy h of the orbiting stars per unit reduced stellar mass in one dimension
is given by
ẋ2 M
−
.
2
x
A substitution of Eq. 2.22 with ẋ = x′ /x according to Eq. 2.19 yields
h=
x′′ = 2hx + M,
(2.22)
(2.23)
which is a regular differential equation of motion of a displaced harmonic oscillator, where
the singularity is removed. By a coordinate transformation of u2 = x together with Eq. 2.22
this can be simplified to
36
2.2 Modification of nbody6
1
u′′ = hu.
2
(2.24)
For two-dimensional coordinate systems these transformations exploit a mapping in the
complex plane by a 2 × 2 matrix, a so-called Levi-Civita matrix. However, a generalisation
in three-dimensions has been achieved by Kustaanheimo & Stiefel (1965), who used a
4 × 4 matrix and included a fourth fictitious coordinate and corresponding velocity. The
advantage of the regularised equations of motion is that a significantly reduced number
of integration time steps is required while the numerical stability of even circular orbits is
improved (Aarseth, 2003).
Hardware acceleration
A significant acceleration of numerical simulations can be achieved by a parallelised force
calculation. This can be achieved by either a connection of several Central Processing Units
(CPUs) via network or by using special purpose hardware acceleration. One approach of
the latter are the so-called Gravity Pipe (GRAPE) boards, which contained chips that
were particularly designed for a fast parallelised calculation of gravitational forces. Nowadays, the chips provided in standard consumer graphic cards Graphical Processing Units
(GPUs) become more convenient since they are less expensive. These were originally build
to rapidly render three-dimensional graphics but became accessible for other operations
with vendor supported programming interfaces like Compute Unified Device Architecture
(CUDA), which is used in our case of Nvidia GPUs for the calculation of the regular forces.
Because of the similarity to the GRAPE boards (from the users point of view) programs
optimised for GRAPE can easily be adapted to GPUs.
One requirement for an efficient performance enhancement is a sufficient number of
particles (> 103 ), since otherwise the communication of the processors becomes to time
consuming compared to the computational benefit.
2.2 Modification of nbody6
2.2.1 Identification of encounter events
The nbody6 code as such does not automatically provide information about the stellar interaction dynamics. In our work group the code nbody6 has been extended by a definition
to identify an encounter scenario. Moreover, a concept to determine the encounter properties like periastron, masses, eccentricity and the time the encounter periastron passage
happened have been implemented.
37
2 Methods
The neighbour scheme of the nbody6 code is used to identify the indices (and therefore
mass ratio) of each encounter event by searching for the strongest perturbing star amongst
the stellar neighbours. An exception are the KS-regularised pairs which are directly treated
as strongest perturbers to each other. Since the individual star-disc encounter simulations
are restricted to two-body perturbations, multiple encounter events are not considered,
which might lead to an underestimation of the encounter influences on the disc properties.
An encounter is identified by the trajectory of the encounter being a nearly Keplerian
orbit of concave shape and second, at least 10% of the orbit have to be fulfilled. If these
requirements are achieved, the periastron distance is calculated. Since the simulation time
is discretised the obtained closest separation is not necessarily the periastron distance (see
Fig. 2.4). Especially in case of strong encounter events with small periods this becomes an
issue. In polar coordinates the separation r can be interpolated as a Keplerian orbit by
~rmin
~rmin
~rp
∆ϕ ~rp
~vmin
Figure 2.4: Sketch of an encounter taken from Olczak et al. (2012). The black circle indicates the position of the
central star while the grey circles mark the position of the perturber at sequential time steps.
r=
(1 + ǫ)rperi
1 + ǫ cos(φ)
(2.25)
with eccentricity ǫ and periastron distance rperi (see Fig. 2.4). Note, that ǫ has not to be
strictly equal in two different time steps due to numerical limitations. Accordingly, slight
differences in the eccentricities are accounted for while for differences of more than 10%
the encounter is rejected.
38
2.2 Modification of nbody6
The periastron distance rperi and eccentricity ǫ are obtained from the Laplace-Runge-Lenz
vector
~ = ~v × ~l − GM ~r,
A
r
(2.26)
where M = m1 + m2 is the total mass of the encountering stars and ~l = ~r × ~v defines the
~ and ~l, are conserved for an inverse-square central
specific angular momentum. Both, A
force. From ~rmin and ~vmin , which are illustrated in Figure 2.4, we obtain
rperi =
and
ǫ=
|~l|2
GM µ
~
|A|
GM
(2.27)
(2.28)
where µ = m1 m2 /M is the reduced mass.
To determine the time of the periastron passage, tperi = tmin + ∆t, one eliminates rperi in
Eq. 2.25 and uses the conservation of the specific angular momentum l = r2 φ̇ once again
to obtain
∆t =
2 (1 + ǫ)2
rperi
l
Z
∆φ
0
1
dφ.
(1 + ǫ cos φ)2
(2.29)
In the following, a case differentiation for ǫ = 1, ǫ < 1, and ǫ > 1 has to be performed.
Forming binaries, given by ǫ < 1, are found to be transient systems that in most cases
remain bound over a small number of orbits. However, around 1/4 of all stars undergo
repeated fly-bys by a companion star. Here, simply the most disruptive encounter, which
means the smallest periastron distance and eccentricity, is recorded, since the focus is on
the relevance of stellar encounters. This might underestimate the disc losses but the effect
will be discussed for the respective cases.
Finally, the result of the tracking procedure is a list of encounter parameters for each
particle that contains information about the time, periastron distance, mass ratio and
eccentricity of the encounter.
2.2.2 Instantaneous gas expulsion
As already outlined in Section 1.5, embedded clusters expel their remnant gas quickly
and the the star formation process ceases. In Chapter 5 cluster models with focus on the
subsequent dynamical cluster evolution will be investigated. In this case the cluster is set
up under the condition of a recent expulsion of the remnant gas. A simple way to mimic
such an instantaneous expulsion process is to adapt the virial ratio Q = T /|W |, where T is
39
2 Methods
the kinetic energy, and W the potential energy, to the modified total energy of the cluster,
which has been intensively discussed in Goodwin (1997).
Such simplifications avoid the actual modulation of the natal gas since simply the stellar
velocity distribution has to be adjusted according to the virial state. The correction term
for the velocities is simply obtained by a comparison of the total cluster energies. Here,
the cluster energy before gas expulsion is given by
2
2
2
1
GMtot
GMtot
GMtot
2
E = T + W = Mtot σ 2 −
−
= Q0 Mtot σeq
=
(Q0 − 1) ,
2
2R0
2R0
2R0
(2.30)
2
where Mtot is the total initial cloud mass, R0 is the average radius of the cluster, σ 2 = 2Q0 σeq
2 = |W |/M
is the velocity dispersion of the stellar cluster members, and σeq
tot = GMtot /2R0
is the velocity dispersion if the cluster is in virial equilibrium (Q = 0.5) with the initial
virial ratio Q0 . It can be seen that the cluster is initially unbound (E > 0) if the virial
ratio is Q0 > 1.
However, if the gas is expelled the total cluster mass decreases which results in a lower
total energy after the gas loss, given by
2
2
2
Gǫ2 Mtot
Gǫ2 Mtot
GǫMtot
1
2
−
= Q0 ǫMtot σeq
=
(Q0 − ǫ) ,
E = ǫMtot v 2 −
2
2R0
2R0
2R0
(2.31)
with star formation efficiency ǫ = Mstars /Mtot . The appendant virial ratio given by
Q=
Q0
ǫ
(2.32)
specifies the modified situation of the cluster.
Equation 2.32 has been implemented in the nbody6 code as a correction term for the
initial virial ratio. Thus, the cluster can be modelled as initially out of virial equilibrium
by simply supplying the global star formation efficiency.
In contrast to the instantaneous gas expulsion model, which on the evolution of massive
clusters after the expulsion of the gas, two different approaches will be shortly discussed in
Appendix D.
2.3 Cluster setup
The particles representing the cluster stars in the simulation have to be assigned initial
masses, positions, and velocities. The general setup of the particles will be introduced in
the following.
40
2.3 Cluster setup
Initial mass function
Here, we tried to model a realistic distribution of stellar masses in the cluster instead of
the often used single-mass distribution. Despite the higher computational effort the cluster
evolution is largely determined by the initial mass function (IMF). Hereby, the IMF, ξ(m),
describes the number of stars per unit mass with ξ(m) dm giving the number of single stars
in the mass interval m to m + dm. Thus, the total number of stars with masses between
m1 and m2 is obtained by
N (m1 , m2 ) =
Z
m2
ξ(m)dm.
(2.33)
m1
In contrast to the present-day mass function (PDMF), the IMF describes the distribution
of stellar masses before they start losing mass by outflows or are even completely destroyed
in a supernovae explosion. Therefore, all used IMF’s have been found empirically from
observed distributions of stellar masses in young clusters. Salpeter (1955) found a first
estimate of the initial mass function for the mass range 0.4 < m/ M⊙ < 10, of the form
ξ(m) ∝ m−2.35
(2.34)
indicating that massive stars in stellar clusters are rare in number. While Salpeter’s approach could be confirmed for high-mass stars by later observations (e.g. Massey & Hunter,
1998) it had to be modified for the low-mass regime, where a flattening of the distribution
slope is observed. Currently used IMFs covering the low-mass regime are that by Chabrier
(2003b), who assumes a log-normal distribution for stellar masses m ≤ 1 M⊙ , given by
ξ(log m)m≤1
(log m − log 0.079)2
= 0.158 × exp −
2 · 0.692
(log M⊙ )−1 pc−3
(2.35)
(see also Chabrier, 2003a), and that by Kroupa (2001), given by a broken power law
distribution function



m−0.3


ξ(m) ∝ m−1.3



m−2.3
0.01 ≤ m/ M⊙ < 0.08
0.08 ≤ m/ M⊙ < 0.5
(2.36)
0.5 ≤ m/ M⊙ .
However, due to their low luminosities low-mass stars are difficult to find, which might lead
to an underestimation in this regime.
Throughout this study, the latter approach will be used. Furthermore, here the lower
limit for the stellar mass has been set to the hydrogen burning limit of mmin = 0.08 M⊙
making the plain Kroupa (2001) IMF a sufficient distribution function for this study. Stellar
masses of mstar < 0.08 M⊙ are neglected since they contribute only little to the stellar
41
2 Methods
dynamics and the distribution function is apparently continuous but flattens across the
hydrogen burning limit (see Eq. 2.36 and e.g. Allen et al., 2005). The maximum stellar
mass is set to the accepted upper mass limit mmax = 150 M⊙ (Figer, 2005; Oey & Clarke,
2005; Weidner & Kroupa, 2006). Even though higher stellar masses of 165 − 320 M⊙ have
been recently suggested for massive starburst clusters like R136 (Crowther et al., 2010),
many of such massive stars turned out to be unresolved binaries (Damineli et al., 2000)
(see also Zinnecker & Yorke, 2007, for a review).
Stellar number density distribution
A characteristic quantity, largely determining star-disc encounters in a cluster, is the stellar
number density distribution. In this study, different evolutionary time steps of massive
stellar population are investigated, which require a sophisticated setup of individual density
profiles dependent on its prior formation history.
For the setup of the embedded cluster models (Section 4) the stars have been spatially
distributed according to a density that is representative for an initial ONC density distribution. Observations of today’s ONC density distribution show that it can be approximated
by an isothermal profile (∝ r−2 ) in the outer parts (McCaughrean & Stauffer, 1994; Hillenbrand & Hartmann, 1998) and a flat stellar density profile of the form ρcore ∝ r−0.5 in
the cluster core (Scally et al., 2005).
Since the initial distribution of gas in molecular clouds, which are the birthplaces of
the forming stars (Section 1.1), can be assumed to be isothermal (Larson, 2003), it seems
reasonable to start out with a r−2 dependence for the initial stellar distribution of very
young, embedded star clusters (Scally et al., 2005; Olczak et al., 2006, 2010). Here, an
isothermal density model with an initially slightly increased density in the cluster core
region is used, given by



ρ · r−2.3

 0
ρinitial (r) = ρ0 · r−2.0




0
r ≤ Rcore
Rcore < r ≤ Rcluster
(2.37)
Rcluster < r
where ρ0 = 3.1 × 102 pc−3 , Rcore = 0.2 pc, and Rcluster = 2.5 pc. It is a reliable initial
distribution since it turns out, that after 1 Myr of dynamical evolution, the estimated age
of the ONC, the model matches observations of today’s ONC density distribution (Olczak
et al., 2006) reasonably well.
While starting out with an isothermal stellar distribution seems to be a good approach
to set up very young embedded star clusters (Scally et al., 2005; Olczak et al., 2006, 2010),
the stellar density distribution in the core of more dynamically evolved clusters is found to
42
2.3 Cluster setup
be much flatter (nearly constant) than given by an isothermal sphere.
Thus, for the setup of evolved clusters prior to the expansion phase (Section 5), the stellar
content is distributed according to a King profile of W0 = 9. In contrast to the Plummer
model, which is one of the most commonly used stellar profiles for numerical investigations
due to its analytical simplicity (e.g. Aarseth et al., 1974; Kroupa et al., 2001; Goodwin
& Bastian, 2006; Bastian & Goodwin, 2006; Baumgardt & Kroupa, 2007; Allison et al.,
2009; Brasser et al., 2012), the family of King models provides not only good fits to many
observed old clusters, like globular clusters, but are also applicable for young embedded
clusters like the present ONC (W0 = 9, Hillenbrand & Hartmann, 1998) or even young
starburst clusters like Arches (Initially: W0 = 3† , Present-day: W0 = 7, Harfst et al.,
2010) and NGC 3603 (W0 = 8‡ , Nürnberger & Petr-Gotzens, 2002; Sung & Bessell, 2004),
although their global relation to young massive clusters remains unclear (Portegies Zwart
et al., 2010). In general, King models with W0 ≥ 7 can be denominated as characteristically
for young clusters, due to their concentrated but flattened core.
volume density [M⊙ /pc3 ]
106
King, W0 = 9, rhm = 1.3pc
105
104
1000
100
10
0.1
1
radius [pc]
Figure 2.5: Shown is the cluster (mass) volume density as a function of the distance to the cluster center. A Kingshaped initial distribution profile with W0 = 9, half-mass radius rhm = 1.3 pc (King radius r0 = 0.13 pc),
and containing 30 000 stars is shown in solid red
The King profile of W0 = 9 is shown in Figure 2.5, where the cluster volume density is
†
Private communication with S. Harfst and C. Olczak revealed that an initial King Parameter of W0 = 3
for the Arches Cluster only leads to slightly improved profiles than initial parameters of 3 < W0 ≤ 7.
Higher King parameters of W0 > 7 were not even tested but seem in a first effort also very likely, especially
when considering primordial mass segregation.
‡
Nürnberger & Petr-Gotzens (2002) find a concentration of c = 1.75 which can be converted to W0 ≈ 8
according to Figure C.1. Sung & Bessell (2004) also mention that the King profile fitting is very sensitive
to the central density, which easily leads to a potential underestimation of W0 .
43
2 Methods
plotted as a function of the distance to the cluster center. The red line indicates the Kingshaped distribution given for 30 000 stellar members and half-mass radius rhm = 1.3 pc
(King radius r0 = 0.13 pc, Eq. C.6) as it will be used in Section 5. While the density
distribution continuously decreases towards the cluster outskirts, it flattens for the inner
0.1 pc (see Sec. 4.2 for dynamical evolution of a stellar cluster).
Apart from providing suitable density profiles for evolved clusters, it turns out that King
models also fit the nearly isothermal profile used to set up embedded clusters (Eq. 2.37)
reasonably well assuming a King Parameter of W0 = 12. Figure 2.6 shows the isothermal
volume density profiles according to Equation 2.37 (black lines) and the King W0 = 12
profile (red lines). It can be seen that the initial profiles (solid lines) as well as the evolved
profiles after 1 Myr (dashed lines) are consistent with each other. Only minor deviations
are given due to the sharp edge of the isothermal distribution in the outer cluster parts
(r = 2.5 pc).
volume density [M⊙ /pc3 ]
106
Isothermal
King, W0 = 12
105
104
1000
100
10
0.1
1
radius [pc]
Figure 2.6: Shown is the cluster (mass) volume density as a function of the distance to the cluster center. A Kingshaped initial distribution profile with W0 = 12, rhm = 1.3 pc, and containing 30 000 stars is shown in
red and a nearly isothermal distribution (Eq. 2.37) in black. The solid lines present the initial and the
dashed lines the profiles after 1 Myr.
Having modelled a stellar density profile, all stars are distributed as being initially single,
no primordial binaries are included. This approach will presumably underestimate the
number of ejections from the cluster, since apart from an increased destruction rate of the
stellar discs in tight binaries, the cluster dynamics are usually influenced by strong few-body
interactions (Hills, 1975; Heggie, 1975). Actually, the binary fraction in the ONC today is
found to be around 50% (Duquennoy & Mayor, 1991; Köhler et al., 2006). The primordial
binary frequency could be even higher (≈ 74% Kaczmarek et al., 2011) as dynamical
interactions can destroy wide binaries very quickly. Hence, excluding primordial binaries
44
2.3 Cluster setup
presumably has a non-negligible effect, so that further studies are needed to quantify the
influence on the disc destruction rates. However, binaries are not only destructed but also
form during the cluster evolution, especially if there exist no primordial binaries (Pfalzner &
Olczak, 2008). Here, such evolutionary binary formations are implemented in the encounter
statistics by considering the most disrupting fly-by (Section 2.2.1). Primordial binaries have
been neglected due to the high computational effort but might be an improvement for future
investigation.
Velocity distribution
After distributing the stars within the cluster and assigning masses to them, the last parameter to be attached to the stars is their velocity. The motion of a large number of stars
in a cluster can be approximated by that of molecules in a gas. Hereby, the bodies are
considered as point masses interacting via point center forces. However, these forces differ
considerably. While the forces between gas molecules are rather violent, being significant
on short distances and timescales, the motion of stars in a cluster is influenced by the mean
potential generated by the overall cluster density structure (for more details see Binney &
Tremaine, 2008). Assuming a cluster consisting of 104 stellar members and a crossing time
in the order of 105 yr, physical collisions between two interacting stars become significant
on time scales of the order 107 yr (Eq. 1.8). Consequently, for the short-term evolution
the stellar clusters can be approximated as collision-less systems, which here means only
elastic interactions are considered like in case of an ideal gas.
The velocity distribution of molecules in a thermalised ideal gas is well-described by a
Maxwellian velocity distribution given by
~v 2
f (~v ) = N exp − 2
2σ
,
(2.38)
where f (~v )dvx dvy dvz is the probability of a particle with velocity between ~v and ~v + d~v ,
N the number of particles, and σ the velocity dispersion. For an isothermal sphere σ is
constant throughout the sphere. Moreover, from Equation 2.38 follows that
f (~v ) ∝ f (vx )f (vy )f (vz ).
(2.39)
Here, the Maxwell distribution is obtained by a random number sampling of the Gaussian
deviates for each velocity component. A convenient method for the setup is described in
Box & Muller (1958). First, two random numbers, x1 and x2 , that are uniformly distributed
in the interval [0, 1] are generated and then used for a transformation of the form
y1 =
p
−2 ln x1 cos(2πx2 )
(2.40)
45
2 Methods
y2 =
p
−2 ln x1 sin(2πx2 )
(2.41)
which have a Gaussian distribution with a standard deviation of one. A numerically
faster and more robust calculation is given if polar coordinates (y1 = r cos φ, y2 = r sin φ)
p
p
are assumed, which results in r = y12 + y22 = (−2 ln x1 ) and φ = 2πx2 .
Statistical robustness
It is indispensable to perform multiple simulations (Nsim ) for each individual cluster model
to obtain statistically robust results. Therefore, a set of random initial stellar positions,
velocities and masses has been achieved for each run, which are analysed and averaged
in a subsequent step. It turned out that tracking about 500 000 stellar trajectories leads
to an error of < 3% for the present results. This means for an initial number of stars
Nstar = 1 000 around 500 simulations were preformed while around 15 simulations were
performed in case of Nstar = 30 000. However, due to an inhomogeneous distribution of
stars as well as single stellar masses within the cluster certain parameter sub-samples like
the high-mass regime might lead to slightly increased statistical variations, which will be
discussed in the respective cases.
2.4 Diagnostic aspects
The final step of the data analysation is a combination of the results from the star-disc
encounter parameter study and the dynamical cluster simulations. This will be done in the
diagnostic step. Two approaches have been implemented to merge the parameter study
results with the dynamical encounter parameters recorded from the cluster simulations,
like the minimum stellar separations, the stellar mass ratios and the eccentricities: (i) An
approximation of the disc losses by a fit formula (as provided in Section 3.2.4 - see Eq. 3.4
and Eq. 3.4) or (ii) a first-order interpolation of the tabulated individual disc loss results.
The fitting formula is strictly valid just in a limited mass ratio regime m2 /m1 < 20 as will
be discussed in detail in Section 3.2.4. However, an interpolation method has been preferred
in this study since it significantly improves the loss estimates for the large parameter
domain, in particular for high mass ratios. Therefore, a two-dimensional interpolation
of first order for the discrete mass ratio and periastron quantities, respectively, has been
implemented that generalise the losses from the tabulated parameter study results.
Another important parameter which yet has to be included is the eccentricity ǫ. Here,
a fit formula for the dependence on the eccentricity has been adapted from Olczak et al.
(2010):
46
2.4 Diagnostic aspects
∆x
∆x0
=
· exp (−0.12(ǫ − 1)) · (0.83 − 0.015(ǫ − 1) + 0.17 exp (0.1(ǫ − 1)))
x
x0
(2.42)
where ∆x0 /x0 is the disc mass (or angular momentum) loss obtained by either Eq. 3.4,
Eq. 3.4 or interpolating the single encounter results. ∆x/x gives the final disc loss in
dependence on the eccentricity. Equation 2.42 has been proven valid for varying initial
disc-mass distribution results. It has been tested for the mass losses by averaging over
several simulations with different seeds for the initial particle distribution. The errors lie
typically in the range of 4 − 5 %.
Since parabolic encounters lead to maximum losses, accounting for the eccentricity will
in general reduce the overall disc losses.
47
2 Methods
48
3 Disc-mass distribution in star-disc
encounters
In this part of the thesis the focus is on the effect of the initial disc-mass distribution in single
star-disc encounters between two interacting stars, where one star is initially surrounded
by a circumstellar disc. A parameter study has been carried out in the framework of
my diploma thesis and has been extended here to an increased parameter range to cover
below investigated stellar cluster properties. First, a brief introduction into the numerical
methods and parameter range used in this study will be given. Afterwards the results of the
simulations will be presented including a fit formula for the mass and angular momentum
loss depending on the initial disc-mass distribution index p, followed by a summary and
discussion.
3.1 Setup and method
The encounter between a disc-surrounded star with a secondary star is modelled using
a tree algorithm (see Appendix B and also Pfalzner, 2003). In the here used simulations,
only one star is initially surrounded by a disc. However, previous investigations have shown
that star-disc encounters can be generalised to disc-disc encounters as long as there is no
significant mass exchange between the discs (Pfalzner et al., 2005a). In the case of a mass
exchange, the discs can be to some extent replenished, such that for very close encounters
the mass loss determined in this study would be slightly overestimated.
The disc is represented by 10 000 pseudo-particles, distributed according to a given particle distribution ∝ r−b . Choosing this relatively small number of simulation particles is
motivated by the aim to cover a large encounter parameter space. However, performing test
simulations with 50 000 particles shows that the lower resolution is sufficient for measuring
the properties investigated here.
Initially, the simulation particles move on Keplerian orbits around the central star. The
disc extends from an inner gap of 10 AU to 100 AU. The inner cut-off avoids additional
complex calculations of direct star-disc interactions and saves computation time. Any
pseudo-particle that reaches a sphere of 1 AU around the central star is removed from the
simulation and stated in a commonly used simplified approach as having been accreted onto
3 Disc-mass distribution in star-disc encounters
variable
values
Outer disc radius, rdisc
100 AU
Inner disc radius, ri,disc
10 AU
Number of particles, npart
10 000 (50 000)
Disc mass, mdisc
10−4 M⊙ (10−3 M⊙ )
Particle distribution index, b
0, ( 74 )
Mass distribution index, p
0, 12 , 1,
Rel. perturber mass, M2 /M1
0.1 − 1875
Rel. periastron distance, rperi /rdisc
7
4
0.1 − 50
Eccentricity, ǫ
1
Simulation time, tend
∼ 3 000 yrs
Table 3.1: Initial conditions for all simulations
the star (Bate et al., 2002; Vorobyov & Basu, 2005; Pfalzner et al., 2008). The vertical
density distribution in the disc follows
z2
ρ(r, z) = ρ0 exp −
2H 2 (r)
,
(3.1)
where ρ0 is the unperturbed mid-plane particle density on the equatorial plane with ρ0 ∝
r−(b+1) , where b is the particle distribution index. H(r) is the vertical half-thickness of the
disc (see Pringle, 1981), which is represented by H(r) = 0.05 r according to a temperature
profile of T = T0 · r−1 with T = 20K at the inner edge of the disc. A summary of the disc
properties can be found in Table 3.1.
The total simulation time of tend ≈ 3 000 yrs for each encounter corresponds to three
orbital periods of the outermost particles in the investigated standard disc with size rdisc =
100 AU around a 1 M⊙ star. This time span was found to be adequate for the calculation
of all relevant quantities.
The discs obtained at the end of the simulations continue to develop in the sense that
taking into account viscous forces, the eccentric orbits of perturbed disc pseudo-particles
would re-circularise on timescales probably in excess of 105 yr. However, bound and unbound material can be clearly distinguished shortly after the encounter.
In this present study, we are interested in protoplanetary discs, which are usually of
low mass compared to protostellar discs (Bate, 2011). Here low-mass discs mean mdisc ≪
0.1M1 , where M1 is the mass of the disc surrounded star. The actual adopted disc-mass
value is 10−4 M1 . However, another case of mdisc = 10−3 M1 was tested and no differences in
50
3.1 Setup and method
the result when including self-gravity in the simulations have been found. In addition, test
simulations including pressure and viscous forces have been performed using a SPH code
(for a description of the code see Pfalzner (2003)), but only found negligible differences
(< 3%) in the results. Therefore, in most of the here used simulations self-gravitation and
viscous forces were neglected in favour of higher computational performance.
The actual values of mass and angular momentum loss induced by the encounters were
obtained by averaging over several simulations with different seeds for the initial particle
distribution. The errors in the mass and angular momentum losses were determined as
the maximum differences between the simulation results. They typically lie in the range of
2 − 3 %.
In addition to these obvious performance benefits, the possibility to neglect the selfgravitation, pressure, and viscous forces has allowed us to optimise the computation time
by treating the disc particles as pseudo-particles without a fixed mass during the simulation.
The standard method for generating the initial disc-mass distribution in the simulations
is to assign the same mass to each pseudo-particle (e.g. Boffin et al., 1998; Pfalzner, 2004;
Olczak et al., 2006). This approach involves the performance of a whole suite of simulations
for each variation in the initial disc-mass distribution. In contrast, here, the different discmass distributions are realised by using a fixed pseudo-particle distribution and assigning
their masses after the simulation process according to the desired density distribution
in the disc. The implementation of this flexible numerical scheme allows us to use one
suite of simulations for any initial disc-mass distribution. Note, that the post-processing
of the particle mass is only possible because of the restriction to low-mass discs, where
self-gravitation and viscous forces can be neglected.
The particle masses are assigned to each pseudo-particle in the diagnostic step to calculate
the encounter-induced losses and final mass distributions, which both depend on the initial
distribution. For the example of a power-law disc-mass distribution with index p, the mass
of a particle mi depends on its initial radial position in the disc ri , the total disc mass
mdisc , the number of pseudo-particles npart , their radial positions rj , and the underlying
power-law particle distribution of index b. Since the number of pseudo-particles is limited,
the discrete particle masses mi are described by

npart
mi = 
X
j=1
−1
rj−p+b 
· mdisc · ri−p+b .
(3.2)
The resulting independence of the mass and the particle distributions allows to improve
the resolution of the simulations by initially placing most of the pseudo-particles where the
interaction between the stars and disc is strongest.
In the present simulations, a constant particle distribution (b = 0) has been adopted.
51
3 Disc-mass distribution in star-disc encounters
This is done because the effects on mass and angular momentum are largest at the outskirts
of the disc. Using a constant pseudo-particle density (b = 0) throughout the disc ensures
a higher resolution in the outer parts, even for steep mass density profiles, than for the
standard approach. In contrast, if one were interested in processes that mainly concern the
inner parts of the disc, such as accretion, it would be more accurate to use not a constant
particle distribution but one that guarantees a high resolution close to the star (b ≥ 1).
The method of a posteriori mapping of the particle masses has been tested against
representative test calculations for different initial pseudo-particle distributions and against
the standard method of fixed pseudo-particle masses. The differences in the results were
negligible, thus justifying the generalisation of the results for a constant particle distribution
(b = 0).
Apart from the density distribution, the outcome of an encounter depends on the relative
periastron distance rperi /rdisc and the mass ratio of the involved stars M2 /M1 , where M2
is the mass of the perturbing star. The parameter space that has been investigated - for a
summary see Table 3.1 - is chosen in such a way that it spans the entire range of encounters
likely to occur in a typical young cluster. The lower limit of the perturber mass ratio is
chosen to be M2 /M1 = 0.1 as for smaller mass ratios the influence of the perturber becomes
insignificant. The upper limit, M2 /M1 = 1875, is determined by the maximum possible
mass ratio expected to occur in a massive star cluster. It corresponds to the ratio between
the hydrogen burning limit, which defines the lowest stellar mass of 0.08 M⊙ , and 150 M⊙ ,
the accepted upper stellar mass limit (Figer, 2005; Oey & Clarke, 2005; Weidner & Kroupa,
2006). The inner edge of the disc marks the lower value of rperi /rdisc = 0.1. The upper
value of rperi /rdisc corresponds to the distance where perturbations become negligible and
therefore depends on the perturber mass.
3.2 Results
Although in principle any mass distribution of circumstellar discs can be studied, four exemplary disc-mass distributions are investigated here in more detail. To cover the entire
range of numerically and observationally determined disc-mass distributions, a constant
mass distribution (p = 0) is considered, representing the lower boundary of expected distributions, and a p = 7/4 distribution, providing an upper limit. Additionally, a p = 1
distribution is investigated for comparison to previous star-disc encounter results and a
p = 1/2 distribution, which is in the range of analytical results for low-mass discs (Shakura
& Sunyaev, 1973; Pringle, 1981) and similar to results found in recent investigations of
magnetised material (see Sec. 1.1).
The present investigation focuses on coplanar, prograde encounters on parabolic orbits
52
3.2 Results
(ǫ = 1) (see example in Fig. 3.1a) . Previous studies have shown that even for clusters as
dense as the inner ONC region, most encounters in star clusters are expected to be close to
parabolic (Larson, 1990; Ostriker, 1994; Olczak et al., 2010). These parabolic encounters
provide the strongest impact of the perturber on the disc, since for higher eccentricities
the perturber only interacts briefly with the star-disc system and is, therefore, unable to
influence the disc significantly. The limitation to a certain orientation is more restricting,
as inclined and retrograde encounters can lead to lower losses in mass (Heller, 1993; Hall
et al., 1996; Clarke & Pringle, 1993). However, Pfalzner et al. (2005b) showed that up to
an inclination of 45◦ the mass loss of the disc is only slightly smaller than for a coplanar
orbit. This means that the results have to be regarded as an upper limit for encounters at
different inclinations.
a)
b)
1.5
1
1
0.5
0
z/rdisc
y/rdisc
0.5
-0.5
-1
0
-0.5
-1.5
-2
-1
-2
-1
0
1
2
x/rdisc
-1
-0.5
0
0.5
1
r/rdisc
Figure 3.1: The dashed line in a) shows the boundaries of the initial disc while the solid red line indicates the
trajectory of a grazing perturber (rperi /rdisc = 1) of equal mass (M2 /M1 = 1). Material that resides
within the disc after the perturbation is marked as black squares, while material that is in the end either
bound to the perturbing star, unbound, or accreted is shown as grey dots. Note, that the simulations
were performed in three dimensions as can be seen in b).
3.2.1 Surface density distribution
To achieve a clearer understanding of the re-distribution of the disc material during a
star-disc encounter that depends on the initial disc-mass distribution, first, the focus is
on the evolution of the surface density profiles. In Fig. 3.2 is presented how the evolution
of the surface density in star-disc encounters depends on the periastron distance for both
a constant initial disc-mass distribution (p = 0) and one with a steep initial distribution
(p = 7/4). It shows the mass distributions Σ(r) before (solid black line) and after a
penetrating (rperi /rdisc ≤ 1, dashed blue line), grazing (rperi /rdisc = 1, dashed red line),
and distant encounter (rperi /rdisc = 3, dashed green line). Here, the relative perturber mass
was chosen to be M2 /M1 = 1.
53
3 Disc-mass distribution in star-disc encounters
1
1
Σ(r)[10−4 ·
Σ(r)[10−4 ·
mdisc
]
AU 2
b)
mdisc
]
AU 2
a)
0.1
0.1
0.5
r/rdisc
1
0.1
0.5
1
r/rdisc
Figure 3.2: Initial (solid black line) and final surface densities in case of a) initially constant distributed disc material
of p = 0 and b) a steep distribution of p = 7/4. In both plots non-penetrating (rperi /rdisc = 3, dashed
green line), grazing (rperi /rdisc = 1, dashed red line), and penetrating encounters (rperi /rdisc = 0.1,
dashed blue line) are plotted for a perturbing star of equal mass (M2 /M1 = 1).
As known from previous investigations (Hall, 1997; Larwood, 1997), an encounter reduces
the density in the outer parts of the disc by transporting some part of the outer disc material
inwards while another fraction of material migrates outwards. The latter might become
both captured by the perturber and unbound if the encounter is strong enough. These
effects become more pronounced for small rperi /rdisc and/or large M2 /M1 . However, the
actual amounts depend on the initial disc-mass distribution. In an equal-mass distant
encounter (rperi /rdisc = 3), most of the perturbed disc material is pushed inside the disc
towards smaller disc radii. In this case, 12.5 % of the total disc mass migrates inwards
for an initial constant disc-mass distribution (p = 0), whereas it is only 6 % for an initial
p = 7/4.
The effects of different initial disc-mass distributions become even more obvious for closer
encounters such as grazing (dashed red lines in Fig. 3.2) or penetrating fly-bys (dashed
blue lines in Fig. 3.2). Here, the migration process and also the differences between the
initial distributions are dominated by material moving outwards. In the case of a grazing
encounter and an initially constant disc-mass distribution, about 60 % of the disc mass
becomes unbound. However, in addition about 10 % of the total disc mass can be found
outside the initial disc radius of 100 AU that is still bound to the central star. In contrast,
for the steep p = 7/4-mass distributions only ∼ 30 % of the disc mass becomes unbound but
again ∼ 10 % is bound but situated outside 100 AU. It can be concluded that the outer disc
material is generally separated from the disc for grazing encounters, while material initially
located inside the disc (r/rdisc ≤ 0.7) is re-distributed but remains bound to the central
star. Hence, prominent differences in encounter-induced disc losses for the investigated
disc-mass distributions are expected for strong perturbations of the outer disc parts. In
54
3.2 Results
penetrating encounters, part of the disc material is pushed further inside the disc resulting
in a higher surface density in the inner disc regions. In extreme cases, the disc loss can
increase to > 90 % so that the final disc structure can no longer be regarded as a disc.
1
Initital disc-mass distribution
Final disc-mass distribution
Σ(r)[10−4 ·
mdisc
AU 2 ]
const.
0.1
r−5/2
0.01
0.001
0.1
0.25
0.5
1
r/rdisc
Figure 3.3: Initial (p = 0, solid black line) and final surface density after an encounter of M2 /M1 = 5.0 and
rperi /rdisc = 0.7 (dotted red line). The dashed straight line represents a slope of p = 5/2.
In nearly all cases, a steepening of the density profile is the general effect of an encounter.
The degree of steepening depends on the mass ratio, periastron distance, and the initial
disc-mass distribution. Fig. 3.3 shows that even initially flat distributed disc-material
(p = 0) can be redistributed into a surface density profile steeper than p ≈ 2.
How is the vertical structure of the disc influenced by an encounter? As only coplanar encounters are investigated here, the answer is: surprisingly little. Fig. 3.1b shows
the vertical particle profile of a strongly perturbed disc after a penetrating encounter of
M2 /M1 = 1. The main effect of the encounter is a decrease in the number of particles
in the outer disc regions. In these parts of the disc, the resolution is relatively low, but
owing to the here used constant particle distribution (b = 0) it is significantly higher than
in previous investigations using r−1 particle distributions in their simulations.
In a few special cases where material is pushed moderately inwards, even partially negative distribution indices as low as p = −1 can be the end product of an encounter (Fig. 3.4).
The effect is most prominent in the inner part of perturbed discs with initially constant distributed disc material. Nonetheless, the outer parts of the disc always have a distribution
with positive index, p > 0.
This could perhaps explain the negative indices observed for some discs. Observations
remain limited to measuring only part of the radial extension of the entire disc. If this were
55
3 Disc-mass distribution in star-disc encounters
to be the range where negative indices prevail, one would wrongly extrapolate negative
indices for the whole disc, while the overall index of the disc would remain positive.
Initital disc-mass distribution
Final disc-mass distribution
mdisc
AU 2 ]
1
Σ(r)[10−4 ·
r+1
const.
0.25
0.35
0.5
r/rdisc
Figure 3.4: Initial (p = 0, solid black line) and final surface density after an encounter of M2 /M1 = 1.0 and
rperi /rdisc = 2.0 (dotted red line). The dashed straight line represents a slope of p = −1.
3.2.2 Relative mass loss
In addition to changing the shape of the disc-mass distributions, star-disc encounters can
remove material from the disc. Fig. 3.5 shows the relative disc-mass loss ∆mrel = (mt=0 −
mtend )/mt=0 for the four initial disc-mass distribution indices p = 0, p = 1/2, p = 1, and
p = 7/4. Two representative cases of orbital parameters are singled out: Fig. 3.5a shows
the dependence on the periastron distance for an equal-mass encounter, whereas Fig. 3.5b
depicts the dependence on the mass ratio for grazing encounters (rperi /rdisc = 1).
Qualitatively, the dependence of the relative disc-mass loss on the periastron distance
agrees with previous results (Pfalzner et al., 2005b; Olczak et al., 2006). However, the absolute values change considerably for the different initial disc-mass distributions. The effect
is largest for nearly grazing encounters where mainly the outer disc regions are affected.
As expected from the respective fraction of material in the outer regions, maximal mass
losses occur for initially constant disc-mass distributions, while minimal losses occur for the
r−7/4 -distribution. The largest difference in mass loss between the investigated disc-mass
distributions in Fig. 3.5a occurs for rperi /rdisc = 0.9, whereas the r−7/4 -distribution only
has a mass loss of 33 %, and the constant mass distribution has a mass loss of 64 %.
The situation is somewhat different for very close penetrating encounters (rperi /rdisc ≤
0.3), where the discs are so strongly perturbed that the resulting structure can hardly be
56
3.2 Results
described as a disc. In this case the disc-mass loss seems relatively independent of the
initial mass distribution (see Fig. 3.5a). At the other end of the parameter space - i.e. at
large relative periastron distances - the mass loss becomes too small (∆mrel ≤ 10%) to
infer any dependence on the initial distribution.
1
b)
1
0.8
0.8
0.6
0.6
∆mrel
∆mrel
a)
0.4
0.2
0.4
0.2
0
0
0.5
1
1.5
2
rperi /rdisc
2.5
3
3.5
4
0.1
1
10
100
M2 /M1
Figure 3.5: The relative disc-mass loss of a p = 0 (black line), p = 1/2 (blue line), p = 1 (red line), and p = 7/4 (green
line) disc-mass distribution including all particles bound more tightly to the central star than to the
perturber and excluding unbound and accreted particles. The data is plotted for a) different periastron
distances and an equal-mass perturber, and b) different perturber mass ratios and rperi /rdisc = 1. The
vertical dashed line indicates the initial outer disc radius.
Similarly, the dependence on the initial mass distribution is less pronounced for M2 /M1 <
0.3 and M2 /M1 > 90 (see Fig. 3.5b for a grazing encounter). Hence, generally weak
perturbations - whether distant or of low mass ratio M2 /M1 - are incapable of significantly
influencing the discs, while in the case of strong perturbations nearly the entire disc material
is removed independently of the investigated disc-mass distributions. In both cases, the
mass loss does not depend strongly on the disc-mass distribution. In contrast, encounters of
intermediate strength are most sensitive to the disc-mass distribution. For the case shown
in Fig. 3.5b, a maximum differences of up to 35 % for M2 /M1 = 3 is determined.
For high-mass ratios M2 /M1 > 20 and certain non-penetrating periastron distances,
which restrict the gravitational star-disc interactions to the outer disc parts, differences in
mass loss as high as 40 % can be inferred for the different initial disc-mass distributions.
Having determined the relative disc-mass loss in individual star-disc encounters, Figure 3.6 shows the losses of an initially constant disc-mass distribution divided by the losses
of a p = 7/4 distribution, as a function of the periastron distance. Three representative
mass ratios are presented, M1 /M2 = 0.1, 20, and 500. It can be seen, that the larger the
periastron distances the higher are the relative differences between the disc-mass losses for
the different initial disc-mass distributions. For perturbations of the outermost disc regions
the losses of initially flat disc-mass distributions are around three times higher than the
losses for initially steep disc-mass distribution profiles.
57
3 Disc-mass distribution in star-disc encounters
M1 /M2 = 0.1
M1 /M2 = 20.0
M1 /M2 = 500.0
∆mconst /∆m7/4
3
2.5
2
1.5
1
0.1
1
10
rperi /rdisc
Figure 3.6: Shown is the relative disc-mass loss of a constant disc-mass distribution (∆mconst ), in units of the loss of
a r−7/4 distribution (∆m7/4 ), as a function of the periastron distance. The data is plotted for perturber
mass ratios of M1 /M2 = 0.1 (black line), 20 (blue line), and 500 (red line). Here, the cases of individual
mass losses ∆m7/4 < 3% have been excluded.
3.2.3 Relative angular momentum loss
The different initial disc-mass distributions do not only influence the disc-mass loss but
also the angular momentum loss. Fig. 3.7a shows the relative angular momentum loss
∆Jrel = (Jt=0 −Jtend )/Jt=0 as a function of the encounter distance for equal-mass encounters
and the four different disc-mass distributions used in this work. As expected from previous
results (e.g. Pfalzner & Olczak, 2007), the general trends in relative angular momentum
loss are quite similar to that of the mass loss (compare Figs. 3.5 and 3.7) with angular
momentum losses being slightly higher than the disc-mass losses.
While the mass losses for encounters of intermediate strength with rperi /rdisc = 0.9 and
M2 /M1 = 1 are 64 % for the constant mass distribution compared to 33 % for the 7/4mass distribution as shown before, the corresponding angular momentum losses are 75 %
and 50 % , respectively. Material migrating inwards or becoming unbound owing to an
encounter leads to an angular momentum loss, while the fraction of the disc mass that
is pushed beyond the initial disc-radius but remains bound to the central star increases
the total angular momentum of the disc. In total, the dependence on the initial discmass distributions is less pronounced for the relative angular momentum loss than for the
disc-mass loss.
A similar result is found for the relative differences between the angular momentum
loss of initially constant (∆Jconst ) and p = 7/4-density distributed discs (∆J7/4 ) shown in
58
3.2 Results
1
b)
1
0.8
0.6
0.6
∆Jrel
0.8
∆Jrel
a)
0.4
0.4
0.2
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0.1
rperi /rdisc
1
10
100
M2 /M1
Figure 3.7: The relative angular momentum loss of a p = 0 (black line), p = 1/2 (blue line), p = 1 (red line),
and p = 7/4 (green line) disc-mass distribution including all particles bound more tightly to the central
star than to the perturber and excluding unbound and accreted particles. The data is plotted for a)
different periastron distances and an equal-mass perturber, and b) different perturber mass ratios and
rperi /rdisc = 1. The vertical dashed line indicates the initial outer disc radius.
M1 /M2 = 0.1
M1 /M2 = 20.0
M1 /M2 = 500.0
∆Jconst /∆J7/4
3
2.5
2
1.5
1
0.1
1
10
rperi /rdisc
Figure 3.8: Shown is the relative angular momentum loss of a constant disc-mass distribution (∆Jconst ), in units of
the loss of a r−7/4 distribution (∆J7/4 ), as a function of the periastron distance. The data is plotted
for perturber mass ratios of M1 /M2 = 0.1 (black line), 20 (blue line), and 500 (red line). Here, the cases
of individual angular momentum losses ∆J7/4 < 3% have been excluded.
59
3 Disc-mass distribution in star-disc encounters
Figure 3.8. Here, perturbations of only the outermost disc regions yield to an up to 2.5
times higher angular momentum loss for the initially p = 0 distribution than obtained for
the case of p = 7/4.
However, although the differences are lower in maximum the influence of different initial
disc-mass distributions on the angular momentum loss covers a significantly large parameter range.
Since mass and angular momentum losses are generally influenced by perturbations of
the outer disc parts, an initially flat particle distribution that has a high resolution in
the outer disc regions can help us to achieve a higher accuracy in determining disc losses.
Nevertheless, disc losses obtained with initially steep particle distributions (p = 1) found in
previous studies (Olczak et al., 2006; Pfalzner & Olczak, 2007) are accurately reproduced
within the error range in the present study.
3.2.4 Adapting a fit formula dependent on the initial disc-mass distribution
The numerical results for the mass and angular momentum loss in this study cover a
wide parameter range but, however, present only a discrete classification of the relative
losses for the different initial disc-mass distributions. Analytical approaches are a possible
option to avoid this disadvantage but are only valid for a very limited parameter space of
distant encounters (Ostriker, 1994; D’Onghia et al., 2010). To obtain a general estimate
of the effect of arbitrary initial disc-mass distributions on the encounter-induced mass and
angular momentum loss of protoplanetary discs, a fit formula is presented that is valid for
any initial disc-mass distribution, given that it can be expressed by a power law of the
form Σ(r) ∝ r−p . For this purpose, the fit function for the relative mass loss of Olczak
et al. (2006) (see Eq.4 therein) and for the relative angular momentum loss of Pfalzner &
Olczak (2007) (see Eq.1 therein), which are valid for a r−1 -distribution, is extended towards
arbitrary surface density distribution indices 0 ≤ p ≤ 7/4
∆mrel =
M2
M2 + 0.5M1
( r
× exp −
∆Jrel
60
1.2
h
i
p
ln (3 − ) · (rp )0.1
4
)
h
i
M1
1−p
2−p/2
|
(rp )
− 0.5 p + |
,
M2 + 0.5M1
2
1 − p exp(p) 0.5·rp
M2
+
−
= 1.02
M1 + M 2
5
100

 s
M1 (rp − 0.7 rp0.5 )3
,
× exp −
M2
(3.3)
(3.4)
3.2 Results
where rp = rperi /rdisc is the relative periastron distance.
For most of the parameter space, the adopted functions fit the data well within the
margin of error and extend Ostriker’s analytical function for the angular momentum loss
considerably. Larger deviations of the fit functions from the simulated losses occur only
for high encounter mass ratios of M2 /M1 > 20, which was also the case for the established fit functions of Pfalzner & Olczak (2007) and Olczak et al. (2006). In the case of
disc-penetrating orbits and low-mass ratios, the disc losses can be affected by Lindblad
and Corotation resonances that are located in the inner disc regions and cause moderate
deviations of the fit function from the expected disc losses.
The extended fit functions provide a significant improvement to previous analytical and
numerical results. They cover an extensive parameter space of the most reasonable initial disc-mass distributions and the most relevant orbital parameter ranges expected for
interactions in star clusters of any age.
3.2.5 Definition of disc-perturbing encounter
In the following chapter, the previously described results for single star-disc encounters will
be combined with simulations of the dynamical evolution of stellar populations. The aim
will be an extensive investigation of the influence of the initial disc-mass distribution on stardisc encounters in the different developmental stages of a dynamical cluster environment.
In this context, a definition for a disc-perturbing encounter is needed.
The analytical identification of a stellar encounter event has already been introduced in
Section 2.2.1. However, not all of such encounters have necessarily a fundamental influence
on the circumstellar discs. To investigate the importance of star-disc encounters in cluster
environments a criterion is required, which classifies a disc-perturbing encounter.
Due to a limited resolution of current telescopes, observations are generally constrained
to the extreme cases of the complete destruction of a disc. A typical observed quantity in
this context is the cluster disc fraction that has been discussed extensively in Section 1.3.
Throughout this study, a star is termed as disc-less and a disc as destroyed if ≤ 10% of its
initial disc mass remains bound to the central star.
However, already non-destructive encounters significantly influence the structure of circumstellar discs. Encounter-induced density fluctuations might even accelerate the planet
formation process via gravitational instabilities within the discs. The relevant physical
quantity here to determine the effect of a stellar fly-by on the disc structure is the angular momentum loss. In Figure 3.9 three perturbed discs are shown for differing encounter
parameters but similar angular momentum losses of ∆Jrel ≈ 5%. In all these cases the
complete disc mass remains bound to the central star. It can be seen that a similar disc
structure is obtained for several encounter mass ratios and periastron distances. A signifi-
61
3 Disc-mass distribution in star-disc encounters
1
y/rdisc
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
0.5
1
1.5
0.5
1
1.5
0.5
1
1.5
x/rdisc
1
y/rdisc
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
x/rdisc
1
y/rdisc
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
x/rdisc
Figure 3.9: Here, three examples of a perturbed star-disc system with an angular momentum loss of ≈ 5% are shown
for a) an encounter of mass ratio M2 /M1 = 0.5 and periastron rperi /rdisc = 2.5, b) M2 /M1 = 1 and
rperi /rdisc = 3.0, and c) M2 /M1 = 90.0 and rperi /rdisc = 11.5. The dashed line shows the boundaries
of the initial disc and the yellow asterisk illustrates the position of the central star.
62
3.3 Conclusion
cant perturbation of the disc outskirts can be determined, accompanied by the formation
of two tidal tails, which are a result of the gravitational attraction of the perturbing star
and the relative movement between the central star and the center of mass of the disc
(Pfalzner, 2003). Only a few observations of such spiral structures exist (e.g. Lin et al.,
2006; Muto et al., 2012), not only because of the limited observational resolution, but
also due to re-circularisation processes, which might lead to a smoothing of the density
fluctuations within a few 104 Myr.
To investigate the overall effect of stellar encounters on circumstellar discs in a stellar
environment, in the following the disc around a star is defined as perturbed if ∆Jrel ≥ 5%.
In general, the number of stars with perturbed discs is higher than the number of stars
with disc-mass loss ∆mrel ≥ 3%. An encounter leading to a perturbed disc is called a
disc-perturbing encounter.
3.3 Conclusion
The circumstellar disc profiles largely determine the shape of the forming planetary systems.
Recently, Chiang & Laughlin (2012) analysed the distribution of extrasolar planets found
by the Kepler mission and suggested a Minimum Mass Extrasolar Nebula with a power-law
index p = 1.6, which is even slightly steeper than the one for the Minimum Mass Solar
Nebula of p = 1.5 (Weidenschilling, 1977). The probably most realistic model for the
formation of the Solar System is the Nice model (Gomes et al., 2005; Tsiganis et al., 2005;
Morbidelli et al., 2005), which describes a dynamical evolution scenario for the formation
of the Solar System. Desch (2007) deduced that according to the Nice model the initial
surface density profile of the solar nebula protoplanetary disc must have been even steeper
with a Σ(r) ∝ r−2.2 form to develop into the present Solar System. As a possible way of
forming such steep disc-mass density profile, he considered photoevaporation by an external
massive star (see also Mitchell & Stewart, 2010). Kuchner (2004) found that similar surface
density profiles are required for the formation of known extrasolar planetary systems. In
general, these results suggest rather steep disc-mass distribution profiles for the potential
formation of planetary systems.
Here, it has been demonstrated that the consequence of an encounter is a steepening in
the surface density of the disc on short timescales with close encounters invoking density
profiles steeper than Σ(r) ∝ r−2 for any given initial disc-mass distribution. Since these
density profiles are probably the prerequisite for forming a planetary system similar to the
Solar System, encounters in the early history of the Solar System might have provided
these conditions (Adams, 2010).
Moreover, the results emphasise that these steep density profiles do not have to exist
63
3 Disc-mass distribution in star-disc encounters
ab initio or be formed by photoevaporation processes but that even discs that initially
have evenly distributed material can fulfil the requirements for the formation of a Solar
System type planetary system in the inner disc regions after a close encounter. Additional
evidence of such an encounter in the case of the early Solar System was given by distant
Solar System objects on highly eccentric orbits, such as the transneptunian object Sedna,
and also the sharp outer edge of the Solar System at ∼ 50 AU from the Sun (Ida et al.,
1999; Allen et al., 2001; Morbidelli & Levison, 2004; Kenyon & Bromley, 2004). Pfalzner
(2012) showed that the Solar System most likely formed in a leaky cluster environment
with an encounter probability of ≈ 30% in the first 2 Myrs.
On the other extreme, in rare cases encounters can not only lead to profile steepening but
also to profile flattening, which might explain even the surface density distribution profiles
of p < 0 observed by Isella et al. (2009).
Consequently, if all young stars were to start out with the same disc-density structure,
the influence of the cluster environment by means of encounters in the very early and dense
phases of cluster evolution could account for the observed multitude of today’s disc-mass
density profiles.
3.4 Summary
As most stars probably form in clustered environments encounters are likely to have a
significant impact on the disc structure. The scope of this part of the study was to determine
the role of the initial mass distribution in this context. The parameter study of star-disc
encounters performed in my diploma thesis was extended for disc-mass distributions of the
form Σ(r) ∝ r−p with p ∈ [0, 1/2, 1, 7/4]. The results can be summarised as follows:
1. The relative disc-mass loss differs by up to 40 % among the different initial density
distributions for the same type of encounter. The largest differences are associated
with perturbations of the outer disc edge, where the mass loss can be up to three
times higher for flat disc-mass distributions (p = 0) compared to steep distributions
(p = 7/4).
2. The dependence on the initial disc-mass distribution is less pronounced for the relative
angular momentum loss in such encounters. Here, differences of 15% between the
investigated distributions are typical.
3. The general effect of an encounter is a steepening of the surface density slope for any
initial disc-mass distribution, which can result in distribution indices of p > 2 even
in the case of initially flat distributions.
64
3.4 Summary
4. The disc-mass and angular momentum losses caused by a parabolic encounter can be
fitted by a function of the perturber mass ratio, the relative periastron distance, and
the index of the initial disc-mass distribution.
In short, the intuitive result that the flatter the mass distribution the stronger the change
in the disc mass and angular momentum caused by encounters has been quantified. This
infers that the importance of encounters in young stellar clusters and their potential to
trigger encounter-induced losses might be reconsidered depending on the dominant initial
disc-mass distribution.
65
3 Disc-mass distribution in star-disc encounters
66
4 Embedded clusters in virial equilibrium
The vast majority of stars seems to be born in massive stellar clusters (Lada & Lada,
2003; Porras et al., 2003; Evans et al., 2009). In the first few Myr of their lifetime the
stars are surrounded by their natal gas, however, the time scale of this embedded period is
observationally not well-constrained, but is expected to be lower than 5 Myr in case of the
most massive clusters (see Section 1.5).
10000
-3
Cluster density [Msun pc ]
-1.3
~rcl
-3
rcl
propagating star formation
exposed clusters
~rcl
-1
100
31.5 r-1
1
0.1
1
10
Cluster radius [pc]
Figure 4.1: Same as Fig. 1.8 but here the simulated density space is indicated by a red bar.
The scope of this section is to investigate this early embedded phase of stellar cluster
development and the general influence on the disc frequencies. One example of such an embedded cluster, where the gas expulsion process near the massive OB-stars recently started,
is the Orion Nebula Cluster (ONC), which has been extensively described in Section 1.4.
Here, it is used as a model cluster since it is one of the best observed young massive star
forming regions that is, at least partly, still embedded in its natal gas. As detailed in Section 1.4, previous numerical investigations found high probabilities for encounters between
the stellar members during the temporal development of such dense stellar cluster environments. Olczak et al. (2010) found that the encounter rate is rather unaffected by the cluster
size but depends strongly on the stellar density. They found that the ONC itself marks
4 Embedded clusters in virial equilibrium
variable
value
number of star
Nstars
4 000
mean stellar mass
mmean
0.5
core radius
rcore
0.3 pc
cluster size
rcluster
2.5 pc
half-mass radius
rhm
cluster core density
ρcore
≈ 1 pc
mean age
tage
crossing time
tcross,hm
2 · 104 pc−3
1 Myr
0.5 Myr
Table 4.1: Properties of the Orion Nebula Cluster (Hillenbrand & Hartmann, 1998, and references therein).
a threshold; in less dense clusters the high-mass stars dominate the interaction dynamics,
whereas in denser clusters the interactions between the low-mass stars themselves play the
major role for the disc-mass loss.
Here, six representative cases of 0.25, 0.5, 1, 2, 4, and 8 times the density of the ONC
have been investigated. The entire range of initial cluster densities covered in this chapter is indicated in Figure 4.1 by a red bar. In addition the observed densities of young
embedded clusters are shown as a function of the cluster size (for more details about the
evolution of the clusters along the sequence shown in black see Section 1.5). The here
covered density range is representative for massive clusters in the late embedded stage.
Clusters of lower densities than considered here might exist as well, but are less prone
to gravitational interactions (Olczak et al., 2010) and, moreover, observationally not well
constrained (Carpenter et al., 2000).
4.1 Setup
The general setup of the stellar masses, radial positions, and velocities has been extensively
described in Section 2.3. However, in this chapter we concentrate on ONC-like cluster models, which involves some assumptions about the setup that are discussed in the following.
The general properties of the ONC are summarized in Table 4.1.
As mentioned in Section 1.4.3, no evident primordial mass segregation scenario could
be confirmed as correct so far. However, Olczak et al. (2011) determined from data of
today’s ONC (Hillenbrand, 1997) a segregation of the five most massive stars towards the
cluster center, while the remaining stars show no signs of mass segregation. Following the
approach by Bonnell & Davies (1998), here, initially the four most massive stars have been
distributed in the inner cluster region with rseg = 0.6 · rhm .
68
4.1 Setup
The stars have been distributed according to Equation 2.37, which describes a nearly
isothermal density profile with a slightly enhanced number density in the core region. To
model the stellar velocities, a Maxwellian velocity distribution (as described in Section 2.3)
with radius-independent velocity dispersion σ has been used. This choice is supported
by observations, as Jones & Walker (1988) found a nearly constant velocity dispersion of
σ1dim ≈ 2km/s independent of the cluster radius by studying stellar proper motions in
√
the ONC. This corresponds to a three-dimensional velocity dispersion of σ = 3 · σ1dim ≥
3.46km/s. Recent observations, investigating radial velocities in the ONC found even higher
values of σ1dim ≈ 3km/s (Fűrész et al., 2008).
As the mean stellar mass mmean ≈ 0.5 (Tab. 4.1), this results in a total cluster mass
MONC = N · mmean = 4 000 · 0.5 M⊙ = 2 000 M⊙ . Assuming above velocity dispersions,
the virial ratio Q is then given by
Q=
rhm σ 2
≥ 0.7,
2GMONC
(4.1)
with the half-mass radius of the ONC given by rhm ≈ 1 pc. From Eq. 4.1 it follows that the
ONC is expected to be in a super-virial state, probably due to recent gas expulsion processes
in the core region. However, observational uncertainties in the parameter determination
(Scally et al., 2005) - and the neglect of low-mass objects (like brown dwarfs) and the gas
component - presumably lead to a considerably underestimation of the total mass MONC ,
which might lead to an overvalued virial ratios. Since in this chapter we are in particular
interested in the effect of stellar encounters rather than the cluster evolution the focus here
will be on initially virialised systems of Q = 0.5. Super-virial clusters will be investigated
in Section 5.
For simulations presented in this chapter, no background potential describing the gas
component is included since the focus is on dynamical snapshots of the stellar cluster before
the gas is expelled. In this case the clusters remain in a roughly virial state. Moreover, after
1 Myr the stellar density profiles match observed embedded clusters (Section 2.3), which
justifies the usage of the present approach for investigations of the relevance of stellar
encounters in young clusters.
The stellar velocities are chosen in such a way as to match an isothermal stellar density
profile in virial equilibrium. The higher mass concentration in the core region (Eq. 2.37)
implies, that in our model the velocities are slightly underestimated which has only a
marginal effect here since the clusters rapidly expand due to the rather short crossing
times.
69
4 Embedded clusters in virial equilibrium
ρcore [103 pc−3 ]
ρcluster [ pc−3 ]
Model
Nstars
Nsim
A
1 000
500
1.3
15.3
B
2 000
250
2.7
30.6
C
4 000
125
5.3
61.1
D
8 000
70
10.5
122.2
E
16 000
32
21.1
244.5
F
32 000
16
42.0
489.2
Table 4.2: The number of initial stars Nstars , number of performed simulations Nsim , number density of the cluster
core region ρcore and total cluster density ρcluster for the six investigated models are shown. The radius
of the core region is rcore = 0.3 pc while the cluster size is rcluster = 2.5 pc. The core density is given by
3
3
ρcore = 3Ncore /4πrcore
and the cluster density by ρcore = 3Nstars /4πrcluster
.
4.1.1 Density scaling
The relative importance of encounters is likely a strong function of the cluster density.
Apart from the ONC-model, density-scaled versions of this model will be investigated to
show the dependence of the results on the cluster density (see also Olczak et al., 2010). As
mentioned before, a suitable range of cluster densities has been modeled that covers the
densities of typical embedded massive clusters (Fig. 4.1).
The density scaling of the star clusters was achieved by keeping the size of the cluster
constant (rcluster = 2.5 pc) and varying the initial number of stars Nstars . Assuming a stellar
number density of roughly ρ(r) = ρ0 r−2 we obtain
Nstars =
Z
rcluster
0
ρ(r)r2 drdΩ ∝ ρ0 rcluster
(4.2)
which shows that in the present case the cluster density scales with the number of stellar
members. As mentioned in Section 2.3 a slightly steeper profile of ρcore (r) ∝ r−2.3 has been
used for the cluster core, however, it has only a minor effect due to the low stellar number
in that region. Here, models with 1 000, 2 000, 4 000, 8 000, 16 000 and 32 000 stars were
investigated, corresponding to cluster densities of 0.25, 0.5, 1, 2, 4 to 8 times the initial
stellar density of the ONC-like models at any radius r < rcluster .
Although the cluster is initially in virial equilibrium, the population is expected to expand
with time due to individual stellar escapers, most likely from the outer cluster regions. The
fraction of escapers is expected to be similar for the density-scaled models, given that the
number of escapers, ∆Nesc , from each shell of width ∆r at radius r from the cluster center
scales with the total number of stars in that shell ∆Nstars . From Eq. 4.3 one obtains
∆Nesc (r) ∝ ∆Nstars (r) ∝ ρ0 ∆r ∝ Nstars ∆r,
70
(4.3)
4.2 General cluster dynamics
so that the fraction of escapers remains constant ∆Nesc (r)/Nstars for the density-scaled
models. Thus, it is expected, that the temporal evolution of the density profiles will be
roughly the same.
An overview of the parameters of the simulated models, like initial number of stars
Nstars and corresponding number of simulations Nsim , is shown in Table 4.2, as well as
the resulting densities in the cluster core region ρcore (with rcore = 0.3 pc, the total radius
of the Trapezium cluster) and the mean cluster density ρcluster within the cluster radius
rcluster = 2.5 pc.
4.2 General cluster dynamics
The parameters that largely determine the dynamical evolution of a stellar cluster are the
virial ratio and the cluster density distribution. In this part of the study, the clusters have
been set up in virial equilibrium, Q = 0.5, which means they are initially neither expanding
nor contracting.
0.0 Myr
1.0 Myr
stellar volume density [M⊙ /pc3 ]
106
105
104
103
102
101
0.1
1
radius [pc]
Figure 4.2: Shown is the stellar mass volume density as a function of the distance to the cluster center for the initial
distribution (blue) and after 1 Myr (red). The results are plotted for model A (dotted line), model C
(dashed line), and model E (solid line).
In Figure 4.2 the evolution of the stellar mass volume density distribution is shown as
a function of the distance to the cluster center. Three representative models are show,
model A (dotted line), model C (dashed line), and model E (solid line). It can be seen
that the initial distributions, shown in blue, are equidistant distributed in accordance with
differences in the initial densities of a factor of four. As expected the density is uniformly
71
4 Embedded clusters in virial equilibrium
increased for a higher number of stars independent of the radial distance to the center.
Due to the primordial mass-segregated setup of the clusters, which involves a subsequent
positioning of the four most massive stars in the cluster center, the densities are here
slightly increased. This effect is most prominent for the sparse clusters where only a minor
fraction of stars is located in the inner regions. However, the evolution of the profiles is
not influenced by this difference as can be deduced from the similarly shaped distributions
after 1 Myr, shown in red. As expected from the analytical predictions in Section 4.1.1, for
all shown cluster models a drop of the inner density can be deduced due to an expansion
of the cluster induced by the stellar escapers.
5%
10%
20%
30%
50%
70%
10
90%
b)
distance to cluster center [pc]
distance to cluster center [pc]
a)
1
0.1
0.1
1
time [Myr]
5%
10%
20%
30%
50%
70%
90%
10
1
0.1
0.1
1
time [Myr]
Figure 4.3: Shown are the Lagrangian radii as a function of the simulation time for (a) model A and (b) model E.
The Lagrangian radii are presented for enclosed mass fractions of 5%, 10%, 20%, 30%, 50%, 70%, and
90%.
The evolution of the stellar volume density over time is demonstrated in Figure 4.3 for
sparse (model A shown in Fig. 4.3a) and dense (model E shown in Fig. 4.3b) clusters. Here,
the Lagrangian radii, representing volumes that include a constant fraction of stellar mass,
are plotted as a function of the simulation time. For model A after 0.1 Myr an expansion
of the core can be seen by an increase of the 5% Lagrangian radius. The relatively short
time scales for the expansion of the innermost cluster regions are in accordance with a
decrease of the crossing time with radius (Eq. 1.7). With time the density wave propagates
outwards and the cluster expands until a new virial equilibrium state is reached.
For model E, shown in Figure 4.3b, which contains a much higher number of stars within
the same volume, the crossing time of the system is significantly reduced which leads to
a much faster expansion of the core region. Moreover, after 3 Myr a deceleration of the
process indicated by a flattening of the 70% Lagrangian radius can be seen. This indicates
a dynamical cooling of the inner system, while the outermost mass fraction, indicated by
the orange line, are probably escapers.
72
4.3 Encounter dynamics
4.3 Encounter dynamics
In the following the cluster dynamics are investigated with focus on stellar encounters. In
this chapter, the exclusive case of initially constant disc-mass distributions will be studied,
since the qualitative results for the dynamical evolution are unaffected by the initial discmass distribution.
10000
1000
100
decreasing
10
1
increasing
averaged number of stars per radius bin
Figure 4.4 shows the number of stars as a function of their radial position after 1 Myr and
a green reference line indicating the initial distribution. The maximum standard deviation
is 2% and all data bins with on average less than one star have not been considered in
Figure 4.4. The evolved stellar positions are divided in two groups, stars which retain part
of their discs (red circles) and disc-less stars (blue squares), which lost more than 90%
of their discs mass due to gravitational interactions (see Sec. 3.2.5). In general, after 1
Myr the number of stars in the core region, rcore = 0.3 pc, is significantly reduced as a
substantial number of stars is ejected to large distances. Stars that maintain their discs
are mostly located in the outer cluster regions, but within 10 pc∗ . Even though these stars
might have been part of an encounter, this event was apparently not strong enough to
remove their discs.
0.1
1
r(t = 1Myr) [pc]
10
20
50
Figure 4.4: Shown is the averaged number of stars per radius bin versus the distance to the cluster center after
1 Myr for model E. Red circles indicate stars which maintain > 10% of their initial disc-mass, while
blue squares represent stars that lost their disc due to gravitational interactions. The initial stellar
distribution is indicated by a green solid line. The standard deviation is ≤ 2%.
∗
On average less than one star per radial bin, initially located in the inner 0.3 pc of the cluster, is ejected
to more than 10 pc without loosing its disc. These stars belong to the minor fraction of escaping stars that
are located in the high-velocity tail of the Maxwellian distribution.
73
4 Embedded clusters in virial equilibrium
By contrast, the disc-less stars are prominent in the innermost and the outermost parts
of the population. Inside 0.2 pc even higher numbers of disc-less stars are found than
stars surrounded by discs. Moreover, basically all stars located outside of 10 pc lost their
discs in encounters. In the outermost parts at around 20 pc a pronounced peak indicates
a significant amount of ejected stars with velocities > 20 pc/Myr. This traces a strong
encounter event within the first few 105 years, which destroyed the discs of these stars and
accelerated them strongly towards the cluster outskirts.
r(t = 0 Myr) [pc]
1
outwards
inwards
0.1
0.1
1
10
r(t = 1 Myr) [pc]
Figure 4.5: Shown is the averaged initial radius versus the actual distance to the cluster center for model E. Red
circles indicate stars which maintain > 10% of their initial disc-mass after 1 Myr evolution, while blue
squares represent stars that lost their disc due to gravitational interactions. The initial stellar distribution
is indicated by a black solid reference line. Radial bins with an average of less than one star have not
been considered.
Figure 4.5 shows the initial distance of the stars to the cluster center as a function of
the distance to the cluster center after 1 Myr. Again blue squares represents the stars
that lose their disc and the red circles stars that remain surrounded by a disc. The solid
black reference line divides an inward from an outward movement. It can be seen that,
as expected, after 1 Myr most of the stars have moved far from their initial positions
accompanied by an increase in the cluster size.
The majority of stars that remain surrounded by a disc are initially located outside of
the inner 0.3 pc. By contrast, the stars which already lost their discs due to gravitational
interactions are initially located within the inner 0.3 pc, indicating that strong perturbations
of the discs preferentially occur in the core region of the cluster.
Stars that are initially located within the 0.1 pc from the cluster center most likely lose
their discs early on and are rapidly ejected. These disc-less fraction of the innermost
74
4.4 Effect of the initial disc-mass distribution
stars gains sufficient kinetic energy to be expelled to distances far beyond 10 pc within 1
Myr. The peak of the distribution of the disc-less stars around 2.5 pc marks the temporal
threshold where the ejection process decreases and the remnant cluster population starts
to equilibriate.
4.4 Effect of the initial disc-mass distribution
Investigating the effect of the initial disc-mass distribution in star-disc encounters, we follow
the approach by Olczak et al. (2010), which only investigated initial distributions ∝ r−1 .
They found a strong influence of gravitational interactions on the circumstellar discs in
the central regions for clusters of intermediate stellar density and even sparse clusters due
to gravitational focusing around the massive stars. For dense clusters encounters are even
more important. However, here encounters between low-mass stars are the major disc
destruction process.
In this section, the focus will be on the upper limits for the disc losses, which is achieved
by assuming all encounters being coplanar and parabolic. Moreover, the focus here will
be on the cluster core regions (rcore = 0.3 pc), representing the densest part of the stellar
population. Initially all stars are surrounded by a disc which are assumed to be equally
distributed with rdisc = 150 AU, representing an average value of observed disc sizes. If
not mentioned otherwise, in the following above characteristics are used to obtain general
upper limits for the disc losses. Afterwards, the limits will be extensively discussed.
4.4.1 Dependence on stellar density
Figure 4.6 shows the fraction of destroyed discs, Fdestroyed , after 2 Myr of cluster evolution
as a function of the central cluster densities in the regime 1.3 × 103 and 4.2 × 104 pc−3
(represented by Model A - F ). It can be seen that for initial disc-mass distributions of
r−1 (filled blue circles) central cluster densities of 103 pc−3 lead to around 15% of all
circumstellar discs being destroyed. For volume densities above 104 pc−3 the fraction of
destroyed discs increases considerably to around 50% † . Here, the fraction of destroyed
discs has been fitted by a second order polynomial function, given by
Fdestroyed (ρ) = 71ρ2 + 5.9ρ + 0.135,
(4.4)
†
The fraction of destroyed discs is slightly lower than in Olczak et al. (2010). The reason is that they
used a fit formula for the disc-mass losses that slightly overestimates the losses in case of mass ratios
M2 /M1 > 20. Here, an interpolation algorithm based on an extended parameter study is used as described
in Section 2.2.
75
4 Embedded clusters in virial equilibrium
where the number density ρ has been scaled with ρ = ρcore · 10−6 pc3 . Fdestroyed (ρ) is
indicated by the solid blue line in Figure 4.6. Note, that the fit function is restricted to
core densities in a range of 103 to 4 × 104 pc−3 , which, however, covers the complete range
expected for the massive stellar clusters in the solar neighbourhood.
fraction of destroyed discs after 2 Myr
0.7
0.6
r−1 mass distribution
r−7/4 mass distribution
constant mass distribution
0.5
0.4
0.3
0.2
0.1
0
2.5
5
7.5 10
25
number volume density [ 103 pc−3 ]
Figure 4.6: The fraction of destroyed discs after 2 Myr of evolution is plotted against the stellar number volume
density in the core region (rcore = 0.3 pc). Circles represent the results of an initial r−1 disc-mass
distribution, while crosses show the constant and open squares the r−7/4 distributions. The filled curve
indicates the standard deviation for p = 1. The fits according to Eq. 4.5 are indicated by the blue solid
(p = 1), dashed (p = 0), and dotted (p = 7/4) line.
Using an initial disc-mass distribution that differs from the r−1 case, one observes a
density-dependent deviation of the cluster disc fraction. For low central densities, ρcore ≤
5 × 103 pc−3 , the disc fraction for the different initial distributions are found to differ only
slightly from the r−1 distribution. Here, again ≈ 15% of the discs are destroyed in less than
2 Myr. For dense clusters larger differences in the fraction of destroyed discs are found.
In the case of 4 × 104 pc−3 (Model F ), for an initially constant disc-mass distribution the
majority of circumstellar discs (60%) are destroyed by encounters, whereas for the r−7/4
distribution the fraction is only about 40%. In other words, 50% more discs are destroyed
for an initially constant disc-mass distribution compared to r−7/4 . As to be expected, here,
an initially constant disc-mass distribution provides a maximum for the encounter-induced
disc mass losses.
The higher the cluster density the larger is the difference in the cluster disc fraction for
the different mass distributions. From relation 4.4 it follows that this difference is as well
a function of the disc fraction, given by
76
4.4 Effect of the initial disc-mass distribution
Fdestroyed (ρ, p)
= 1.22 − 0.22p,
Fdestroyed (ρ)
(4.5)
where p is the initial disc-mass distribution index and Fdestroyed (ρ) the fraction of disc-less
stars in case of an initially r−1 distribution (Eq. 4.4), e.g. for densities of 104 pc−3 a fraction
of Fdestroyed (ρ = 104 ) = 21% of all stars lose their discs, while the fraction is increased to
25% for initially constant disc-mass distributions and decreases to 17.5% for the steepest
distributions. In Figure 4.6 the dashed blue line shows the fit function for p = 0 while the
fit for p = 7/4 is indicated by the dotted blue line. The results have also been tested for
p = 1/2, so that Equation 4.5 can be used in the entire interval of p ∈ [0, 7/4].
Apart from investigations of the disc destruction rates, which basically focus on strong
perturbations of the discs, even weak stellar encounters might influence the disc properties,
such as disc mass and the mass distribution within the disc. This processes are of major
importance but observationally difficult to constrain. To evaluate the general influence
of stellar encounters on the disc properties the focus in the following will be on small
variations of the disc’s angular momentum, which has been defined as being characteristic
for a perturbed disc in Section 3.2.5.
fraction of perturbed discs after 2 Myr
1
0.9
r−1 mass distribution
r−7/4 mass distribution
constant mass distribution
0.8
0.7
0.6
0.5
0.4
0.3
0.2
2.5
5
7.5 10
25
number volume density [ 103 pc−3 ]
Figure 4.7: The fraction of stars with perturbed discs after 2 Myr of evolution is plotted against the stellar cluster
density in the core region (rcore = 0.3 pc). Circles represent the results for an initial r−1 disc-mass
distribution, while crosses show the constant and open squares the r−7/4 distributions. The standard
deviation is shown for the r−1 distribution and is of the same order for the other distributions.
Figure 4.7 shows the fraction of stars with perturbed discs after 2 Myr as a function of
the cluster core density. We find that the fraction increases nearly linearly in logarithmic
space from 40% stars with perturbed discs for ρcore = 1.3 × 103 to 95% for central densities
77
4 Embedded clusters in virial equilibrium
of 4.2 × 104 pc−3 . In dense clusters almost every star experiences an encounter event within
the first 2 Myr in the dense inner cluster regions. The fraction of stars with perturbed
discs, Fperturbed (ρ), can be fitted by
Fperturbed (ρ) = 0.4
ρ
1.3 · 103 pc−3
0.25
,
(4.6)
where ρ is the density within the core region of 0.3 pc. The fit function is shown in Figure 4.7
as a solid blue line.‡
Figure 4.7 shows as well the results for a constant initial disc-mass distribution and
a r−7/4 distribution. It can be seen that the fraction of stars with perturbed discs is
basically not influenced by the initial disc-mass distribution. The reason is that for the
vast majority of perturbed discs angular momentum losses ≫ 5% are obtained. Only a
minor fraction of discs are only slightly perturbed in the range < 5%. This means that our
results are applicable to arbitrary initial disc-mass profiles in the range [0; 7/4]. Thus, even
steep distributions, which are believed to be the prerequisite profiles for planet formation
(Desch, 2007; Meru & Bate, 2011), are significantly influenced by encounters.
Keeping in mind, that for the very dense clusters at least 40% of all circumstellar discs
in the core region are destroyed by gravitational interactions leaves ≈ 55% of stars with
a significantly altered disc structure in the cluster core region. This fraction of stars possibly forms planets under conditions that differ significantly from the isolated case. In
general, the fraction of altered (but undestroyed) discs can be obtained by the difference
of Equation 4.6 and Equation 4.5.
Having investigated the extreme cases of a complete destruction of the circumstellar
disc (> 90% disc mass loss) as well as all significant disc perturbations (> 5% angular
momentum loss), the dependence on the loss thresholds is investigated in the following.
Figure 4.8 shows the cumulative fraction of stars as a function of the angular momentum
loss. For dense clusters (Fig. 4.8a) it can be deduced that the higher the angular momentum
losses the larger is the dependence on the initial disc-mass distribution. However, for
sparse clusters (Fig. 4.8b) the dependence is less pronounced. The reason is that large
differences between the investigated initial disc-mass distributions are apparent for strong
perturbations of the outer parts of the circumstellar discs. Such strong perturbations are
given by either individual high-mass encounters or multiple low-mass perturbations (see
Sec. 3.2.2 and Sec. 3.2.3). This will be quantified in the following.
‡
Note, that the standard deviation for sparse clusters is found to be up to 10%, while the deviations
for dense clusters are only < 2%. The reason is that in sparse clusters disc losses are specified by the few
high-mass stars and depend strongly on their position within the cluster. In dense clusters an increasing
number of low-mass stars is involved in encounter events reducing the prominent effect of the high-mass
stars.
78
4.4 Effect of the initial disc-mass distribution
b)
1
0.8
cumulative fraction of stars
cumulative fraction of stars
a)
0.6
0.4
0.2
r−1 mass distribution
r−7/ 4 mass distribution
constant mass distribution
0
0
0.2
0.4
1
r−1 mass distribution
r−7/ 4 mass distribution
constant mass distribution
0.8
0.6
0.4
0.2
0
0.6
0.8
angular momentum loss [%]
1
0
0.2
0.4
0.6
0.8
1
angular momentum loss [%]
Figure 4.8: Shown is the cumulative fraction of stars as a function of the angular momentum loss after 2 Myr.
The results are shown for (a) Model F and (b) Model A. Circles represent the results of an initial r−1
disc-mass distribution, while crosses show the constant and open squares the r −7/4 distributions.
4.4.2 Dependence on stellar mass
Figure 4.9 shows the fraction of disc-less stars as a function of the stellar mass for (a) Model
A and (b) Model F. For low stellar density environments a prominent loss of disc material
surrounding the massive stars is found (Fig. 4.9a). Here, the high-mass stars in the center
act as gravitational focii, causing multiple encounters with the surrounding stars. Due to
the high number of encounters the entire discs of stars more massive than 50 M⊙ are quickly
removed while stars with masses M1 < 10 M⊙ mostly retain some portion of their discs.
For such low- and intermediate-mass stars only a minor fraction is involved in encounters,
which leads to a disc destruction rate of less than 20% in this regime.
For higher cluster densities the favored encounter scenario changes (Fig. 4.9b). Interactions between low-mass stars become increasingly important leading to large discmass losses for stellar masses of M1 < 10 M⊙ . Disc-mass losses of the high-mass stars
(M1 > 30 M⊙ ) are less prominent, however, still ≈ 50% of the massive stars loose their
disc.
How do these results relate to the effect of the initial disc-mass distribution? Figure 4.10
shows the difference between the fraction of destroyed discs for initially constant and r−7/4
disc-mass distributions as a function of the stellar mass. For dense clusters the relevance of
the initial disc-mass distribution significantly depends on the mass of the central star. For
high stellar masses the encounter mass ratios are usually well below M2 /M1 < 0.5, leading
to only minor differences in the disc losses for different disc-mass distributions as has been
shown in Section 3.2.2. Consequently, here the differences in the fraction of disc-less stars
are found to be relatively low (< 10%).
However, for M2 /M1 > 0.5 much larger differences between the investigated disc-mass
distributions have been found, e.g. > 25% in a single encounter event with M2 /M1 = 1.0
79
4 Embedded clusters in virial equilibrium
Fraction of destroyed discs after 2 Myr
a)
initially const disc-mass distribution
initially steep disc-mass distribution
1
0.8
0.6
0.4
0.2
0
0.1
Fraction of destroyed discs after 2 Myr
b)
1
2
4
10
20
50
100
50
100
stellar mass [M⊙ ]
initially const disc-mass distribution
initially steep disc-mass distribution
1
0.8
0.6
0.4
0.2
0
0.1
1
2
4
10
20
stellar mass [M⊙ ]
Figure 4.9: The fraction of destroyed discs after 2 Myr is plotted as a function of the mass of the central star for
(a) Model A and (b) Model F. The results are presented for an initially constant (light blue) and r −7/4
disc-mass distribution (dark blue). The errors are in general ≈ 2% but depend strongly on the mass
range. Due to a lower number of massive stars per cluster the errors increase to 10% in the high-mass
regime.
80
Difference in destroyed discs after 2 Myr
4.4 Effect of the initial disc-mass distribution
0.22
dense star cluster
sparse star cluster
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.1
1
2
4
10
20
50
stellar mass [M⊙ ]
Figure 4.10: Shown is the difference in destroyed discs after 2 Myr between an initially constant and a r−7/4 discmass distribution as a function of the stellar mass. The results are presented for Model A (red crosses)
and Model F (blue circles). The lines represent Bezier curves. Stellar masses > 50 M⊙ have been
excluded due to a low number of stars in these mass regime and errors of up to 10%.
and rperi /rdisc = 1.0. Such large encounter mass ratios are much more likely for low- and
intermediate-mass stars so that the differences between the losses for the different initial
disc-mass distributions increase to up to 20% for stellar masses < 0.5 M⊙ . Since the large
fraction of the stellar cluster members is of low-mass, the differences in the losses are
dominated by the low-mass stars.
As shown before, despite the stellar masses the effect strongly depends on the stellar
density. The largest differences of > 20% are found for low-mass stars in dense cluster
environments, while for sparse clusters the differences are generally below 5% (Figure 4.10).
By contrast, the relevance of the initial disc-mass distribution for high-mass stars is rather
unaffected by the stellar density. The crucial quantity here is the number of encounters per
star. In general, a higher number of encounters provides larger differences in the fraction of
disc-less stars between an initially constant and a r−7/4 disc-mass distribution. However,
as has been shown in Figure 3.6 the significance depends on the encounter mass ratio.
Figure 4.11a shows the averaged number of encounters as a function of the mass of the
central star after 2 Myr for the inner cluster region of sparse clusters. It can be seen, that
low-mass stars, which maintain their disc, experienced usually less than two encounters per
star, while the number of perturbations for disc-less stars has been five times higher. Here,
it can be deduced that the higher the mass of the central star the larger is the number of
encounters. Due to an average of more than 500 encounters per star the discs of the most
massive stars (M1 > 50 M⊙ ) are generally completely destroyed.
81
4 Embedded clusters in virial equilibrium
a)
disc-less
Number of encounters per star
1000
disc-surrounded
100
10
1
0.1
1
100
stellar mass [M⊙ ]
b)
disc-less
1000
Number of encounters per star
10
disc-surrounded
100
10
1
0.1
1
10
100
stellar mass [M⊙ ]
Figure 4.11: The averaged number of encounters per star after 2 Myr of evolution is plotted versus the mass of
the central star for (a) Model A and (b) Model F. The results are shown for the fraction of disc-less
stars (light blue bars) and the fraction of stars that maintain their discs (dark blue bars). A constant
initial disc-mass distribution was used since the number of encounters does only slightly deviate for the
different investigated initial disc-mass distributions.
82
4.5 Discussion
For dense clusters, like Model F, shown in Figure 4.11b, the number of encounters for lowand intermediate-mass stars increases significantly. While all disc-less stars experienced on
average > 25 encounter events, the fraction of encounters of the disc-surrounded stars is
on average one magnitude smaller. For stellar masses M1 > 75 M⊙ gravitational focusing
still leads to encounter rates of > 100 per star.
In summary, in sparse clusters differences in the fraction of destroyed discs due to varying initial disc-mass distributions are negligible. However, for densities ρcore > 104 pc−3
the increasing number of encounters between low-mass stars results in considerably more
discs being destroyed in case of initially flat disc-mass distributions. The disc fraction of
massive stars remains rather unaffected since their discs are either completely destroyed or
consistently perturbed.
4.5 Discussion
The here obtained results represent upper limits for the influence of encounters in the
different stellar environments. The validity of assumptions made will be discussed in the
following.
Although each star in the cluster is assumed to be initially surrounded by a disc, at
the same time only encounters with a disc-less perturbing star have been investigated to
obtain the disc losses. However, Pfalzner et al. (2005a) showed that star-disc encounter
results can be generalised to disc-disc encounters as long as there is no significant mass
exchange between the discs. In case of close encounters the discs might be replenished
to some extend, which would lead to an overestimate of the losses in strong perturbing
encounters.
Another simplification here is the focus on prograde, coplanar encounters. If the cluster is
not highly rotating, an allignment of the disc and the encounter plane seems rather unlikely.
Pfalzner et al. (2005b) showed that, as long as the inclination is not larger than 45 degrees,
the disc losses are only slightly reduced in comparison to a coplanar encounter. Hence, if
the orientations were completely randomly distributed the losses would be overestimated
in 75% of the encounters.
Similarly, it has been assumed that the relative losses remain unchanged in a consecutive
encounter. Pfalzner (2004) showed that for equal-mass perturbers a second encounter
results not in the same absolute but relative losses as found for the first encounter (see
also Tackenberg, 2009). However, since the perturbations also lead to a hardening of the
remaining discs, our treatment of repeated encounters might lead to an overestimate of the
losses.
Additionally, encounters have been generally treated as parabolic (ǫ = 1) throughout
83
4 Embedded clusters in virial equilibrium
this work. Due to the shorter interaction time scales the total disc losses would drop
considerably for hyperbolic encounter orbits (ǫ > 1). In sparse clusters such hyperbolic
encounters can be generally neglected. By contrast, the average eccentricity of the stellar
orbits is higher in case of dense stellar environments, where the massive stars loose their
dominating role as encounter partners (Olczak et al., 2010).
Another significant factor for disc losses is the initial disc size. Small disc sizes lead to
higher relative periastron distances rperi = r/rdisc and therefore lower disc losses in our
calculations. In this context, observations give no clear picture, providing a multitude of
observed disc-mass distributions and sizes. Thus, here, the discs are assumed to have a
radius with rdisc = 150 AU, which is a typical observed value. However, a scaling of the disc
p
size with the mass of the disc-surrounded star by rdisc = 150 AU · M1 [ M⊙ ]/ M⊙ , as it is
obtained if a fixed force of the stars at the discs outer radius is assumed, would be equally
likely. This would result in an increased disc diameter of massive stars (mstar > 1 M⊙ )
while the disc sizes of low-mass stars (mstar < 1 M⊙ ) are significantly reduced. In the
consequence this would lead to a lower fraction of destroyed discs, since the majority of
stars in the cluster is located in the low mass regime.
All of above simplifications might lead to an overestimating of the presented disc losses.
However, some of the applied assumptions potentially lead to an underestimating.
First of all, sub-stellar objects (Mstar < 0.08 M⊙ ) have been excluded in the present
study. In general, the mass ratios in encounters with such low-mass objects are well below
0.1, which implies that the disc losses would be sufficiently small (< 10%). Hence, the
effect should be minor for massive stars. If the encounter is non-penetrating the effect can
be neglected even for low-mass stellar objects.
Furthermore, all encounter processes have been treated as two-body encounters. Umbreit
(2001) showed that multiple-body encounters result in larger disc losses. However, the effect
strongly depends on the mass and periastron distance of the involved stars. For the present
calculations the most destructive encounter has been used to obtain the disc losses, while
the other encounter partners are most likely either distant or less massive, so that their
influence on the losses is less significant.
Primordial binaries have not been included, which might significantly underestimate the
destruction rates of stellar discs in the portion of tight binaries. While it is suggested that
up to 100% of all stars (e.g. Kroupa, 1995, and references therein) might be initially part of
a binary system, it remains unclear how their initial periods are distributed. Assuming the
upper limit case of an initially log-uniform period distribution (e.g. Reipurth et al., 2007) a
fraction of 50% of all stars would have had a companion with a semi-major axis ≤ 100 AU.
This would have a non-negligible effect on the disc fractions and stellar dynamics and
further investigations are needed to give an estimate of the effect.
84
4.6 Summary
Finally, a large fraction of disc-less stars is ejected after an encounter event within the
first few 105 yr with velocities larger than 20 pc/Myr, populating regions of > 20 pc distance from the cluster center. In the context of planet formation such stellar high-velocity
escapers from embedded clusters are expected to show no infrared excess emission and to
be less frequently surrounded by planets. Similar results have been obtained from observational (Hillenbrand, 1997; Lada et al., 2004) and numerical studies that assumed primordial
mass segregated populations (Olczak et al., 2008). However, such high-velocity escapers
are less frequent for primordial non-mass segregated clusters. Here, a first approach showed
that in contrast to evenly distributed clusters, in mass-segregated clusters a lower fraction
of disc-less stars remains in the core region as more stars are ejected due to gravitational
focusing of the high-mass stars. The consequence is an increased fraction of disc-less stars
in the core region of primordial non-mass segregated clusters of up to 10% in the tested
extreme cases.
4.6 Summary
Stellar clusters spanning a large range of densities have been modeled and the influence of
these different environments on the disc fraction has been investigated. The focus was on
the effect of the initial disc-mass distribution. The main results are:
1. Surprisingly, even though the initial disc-mass distribution significantly influences
individual disc losses induced by stellar interactions, the total number of perturbed
discs remains virtually unaffected.
2. Similarly has the initial disc-mass distribution only a minor influence on the fraction
of discs that are completely destroyed by encounters, as long as the cluster density
is low (ρcore < 104 pc−3 ). The reason is that for low-density clusters interactions
with high-mass stars dominate, which are rather unaffected by the initial disc-mass
distribution.
3. By contrast, for very high cluster densities the fraction of destroyed discs depends
to some degree on the initial disc-mass distribution - e.g. 60% of discs are destroyed
assuming initially constant disc-mass distributions while for an initially steep discmass distribution (Σ ∝ r−7/4 ) only 40% are destroyed. Here, interactions between
low-mass stars are more frequent, which show the largest dependencies on the mass
distribution. In general, the fraction of destroyed discs deviates at most ≈ 20% from
an initial r−1 disc-mass distribution.
We find evidence, that independent of the initial distribution of the disc material, almost
all circumstellar discs (95%) in the core region (rcore = 0.3 pc) of dense clusters are signifi-
85
4 Embedded clusters in virial equilibrium
cantly perturbed. However, the prominent influence on the disc survivability around stars
with < 10 M⊙ would have a non-negligible effect on the frequency of sun-like planetary
systems in dense cluster environments.
86
5 Cluster expansion phase
So far, embedded clusters in virial equilibrium have been investigated. But once a sufficient number of stars have been formed within such an embedded cluster, the remnant
surrounding gas is rapidly expelled by stellar winds, ionizing UV radiation of the massive
stars, and first supernovae explosions (Hills, 1980; Kroupa et al., 2001; Fellhauer & Kroupa,
2005; Goodwin & Bastian, 2006; Bastian & Goodwin, 2006; Baumgardt & Kroupa, 2007;
Dale et al., 2012). This clearing up unveils the dense stellar regions of the cluster where
presumably most star-disc encounters take place.
10000
-3
Cluster density [Msun pc ]
-1.3
~rcl
-3
rcl
propagating star formation
exposed clusters
~rcl
-1
100
31.5 r-1
1
0.1
1
10
Cluster radius [pc]
Figure 5.1: Same as Fig. 1.8 but here a red circle indicates the density space, which marks the here investigated
parameter region assuming an initially super-virial cluster state.
In the following, we concentrate on the most massive clusters in the Milky Way, since
these are the only ones where one can observationally determine the entire expansion process after the gas expulsion. The evolution of such clusters has been detailed in Section 1.5.
In general, the picture is that some embedded clusters, indicated by the filled symbols in
Figure 5.1, gain sufficient stellar mass before the gas expulsion to reach the branch of the
leaky clusters, indicated by the open circles in Figure 5.1. Hereby, the leaky clusters provide an age sequence in the radius-density plane with young clusters evolving from the
5 Cluster expansion phase
upper left towards the lower right part within 20 Myr. This evolutionary path, highlighted
in red, denotes the clusters investigated in this chapter.
The process of star formation turns out to be rather inefficient with typical star formation
efficiencies of ≤ 30% for clusters in the solar neighbourhood (see Section 1.5.2). Therefore,
in these clusters the ejection of the residual gas leads to a rapid decrease of the total cluster
mass. The related reduction of the binding energy leads to a lower the escape velocity for
the stars from the cluster and the residual cluster remains in a super-virial state. Stars
becoming unbound from the remnant cluster cause a further depletion of its potential
energy and, therefore, even more stars are expelled. The total fraction of stars that after
≈ 20 Myr remain bound as a cluster depends on the star formation efficiency ǫ, the ratio
of the gas expulsion timescale to the crossing time τexp /tcross , and the cluster density.
If a rapid expulsion is assumed, the stars are not able to adapt their velocities to the
new potential energy of the cluster, which results in a high loss of stellar members and a
further decrease of the total cluster mass. Moreover, the high velocities of the bound stars
will lead to an enlargement of the cluster radius since the stellar kinetic energies will be
converted into potential energy until a new equilibrium state is reached and the expansion
stops. In summary, the cluster expands, loses mass and thus its mass density decreases
with time as indicated in Figure 5.1.
Here, as done by several previous studies (Goodwin, 1997; Goodwin & Bastian, 2006;
Bastian & Goodwin, 2006; Pfalzner & Kaczmarek, 2013), the focus will be on an instantaneous gas expulsion from the clusters which has been specified in Section 2.2.2. Besides
that, so far the effect of gravitational stellar interactions and their impact on circumstellar
discs in such expanding stellar environments has not been investigated.
Pfalzner & Kaczmarek (2013) determined a fraction of around 4 to 7% of stars becoming
unbound due to encounters in such expanded clusters. This indicates that after the gas
expulsion despite the strong decrease of the stellar density encounters might exist.
5.1 Setup and method
The parameters used in this set of simulations are summarized in Table 5.1. The motivation
for choosing these specific parameters is given in the following.
The masses of the youngest massive clusters shown in Figure 5.1 initially lie in a mass
range from about 8 000 to 25 000 M⊙ and an initial cluster radius rinitial < revolved = 5−7 pc,
where revolved is the observed radius found for the leaky clusters with age < 4 Myr. Pfalzner
& Kaczmarek (2013) performed an extensive parameter study of expanding clusters, which
they compared to the observational values. They found the best agreement between their
simulations and the observations for an initial cluster size of rinitial = 1 − 3 pc, total cluster
88
5.1 Setup and method
variable
value
Number of star
Nstar
30 000
Number of simulations
Nsim
15
Half-mass radius
rhm
1.3 pc
King Parameter
W0
9
ǫ
0.3
Initial virial ratio
Q0
0.5
Expulsion time
texp
instantaneous
Expulsion time delay
tdelay
0.0 Myr
Star formation efficiency
Table 5.1: Properties of the leaky cluster models.
masses > 10 000 M⊙ , and a star formation efficiency of ǫ ≈ 0.3. In this case at least half,
but possibly all, massive leaky clusters follow the observed sequence.
Here, a similar parameter study is performed, but this time with the emphasis on the
encounter history. Simulations were performed with an initial stellar mass of 18 000 M⊙ ,
corresponding to 30 000 stellar members, which represents the mean total stellar mass
expected for young leaky clusters and lies well above the lower limit of 10 000 M⊙ (Pfalzner,
2009). The half-mass radius of the cluster is set to rhm = 1.3 pc, which corresponds to the
best fitting models in Pfalzner & Kaczmarek (2013). Assuming a low value for the half-mass
radius seems reasonable since stellar evolution, primordial stellar binaries, and interactions
with an external tidal field have not been taken into account in our study, which might
lead to an underestimation of the cluster expansion after > 10 Myr (see Section 1.5.2 for
more details). In the following, a star formation efficiencies of ǫ = 0.3 will be assumed.
The cases of ǫ = 0.2, ǫ = 0.4, and ǫ = 1.0 have been simulated as well but will only be
mentioned explicitly in the respective cases.
The focus here is on the post-embedded cluster phase after an instantaneous rapid expulsion of the residual gas component. The aim is to investigate the relevance of star-disc
encounters during the induced dynamical expansion of the cluster. In Section 1.5.2 is outlined, that the timescales of the gas expulsion process strongly influence the development of
the emerging remnant cluster. However, gas expulsion is expected to proceed rather quick
(Goodwin, 1997; Fellhauer & Kroupa, 2005), which is why a first adequate estimate is an
instantaneous gas expulsion from the cluster. In the massive clusters that is not only caused
by the high stellar densities but also because the likelihood of massive stars increases. Due
to strong stellar feedback, in particular from the massive stars, the gas ejection phase
in such massive populations is significantly accelerated. Therefore, an instantaneous gas
expulsion is investigated in the following simulations.
89
5 Cluster expansion phase
Furthermore, the clusters are assumed to be in virial equilibrium (Q0 = 0.5) before the
onset of the gas expulsion so that any star formation efficiency ǫ < 1.0 will lead to an
expanding cluster (Sec. 2.2.2). This has been done since it is supposed that gas expulsion
occurs after a few crossing times (tcross = 0.3 Myr) so that the stars had sufficient time to
virialise (Lada et al., 1984).
In contrast to the clusters being in virial equilibrium, as assumed here, a sub-virial
dynamical state of the stellar clusters before gas expulsion has been suggested (Geyer
& Burkert, 2001; Goodwin, 2009; Allison et al., 2009). This would result in a higher
finally bound cluster mass even in case of very low star formation efficiencies. According
to Equation 2.32 it follows that even if an initially contracting stellar population would
be assumed (Q0 < 0.5) it might expand after the expulsion of the gas whenever ǫ <
2Q0 . Furthermore, Equation 2.32 indicates that in case of an instantaneous gas expulsion
the results can be applied to arbitrary initial virial ratios, so that the here termed star
formation efficiency is in fact an effective star formation efficiency (Verschueren & David,
1989; Goodwin & Bastian, 2006), which is a measure of how much the cluster deviates
from virial equilibrium after the remnant gas is expelled. Since in our case the clusters are
initially in virial equilibrium, the effective star formation efficiency is equivalent to the star
formation efficiency. Although only initially virialised clusters will be investigated in the
following, the here presented results are therefore also applicable for observed non-virial
clusters mentioned above.
Since, according to Equation 2.32, the virial ratio after the instantaneous gas loss can be
simply adapted by the star formation efficiency, an initial virial ratio of Q = Q0 /0.3 = 1.67
is obtained in the present case of ǫ = 0.3 and Q0 = 0.5. Accordingly, the initial velocity
p
dispersion σ is a factor of 1/ǫ = 1.83 larger than obtained for virial equilibrium (see
Eq. 1.6).
The stars have initially been distributed according to a King profile of W0 = 9, which
seems to be a reasonable estimate for young massive clusters in the late embedded stage (see
Section 2.3). In previous studies the often used Plummer model generally leads to a strong
decrease in the residual bound stellar mass due to its much flatter distribution in the central
cluster regions. By contrast, the here used King distribution is much more concentrated
in the inner cluster regions making such clusters prone to gravitational interactions of the
stellar members. A detailed characterisation of Plummer and King profiles is presented in
Appendix C.
90
5.2 Cluster survivability
5.2 Cluster survivability
Several studies investigated the fraction of stars that remain bound to their natal cluster
after the gas expulsion process (see e.g. Figure 1.9). Here, a short overview of the cluster
survivability in dependence on the star formation efficiency will be given for the here used
model cluster described above.
In Figure 5.2 the fraction of bound stars is shown as a function of the simulation time
tsim , where tsim = 0 denotes the time step of the instantaneous expulsion of the gas. Note,
that the simulation time is not equal to the cluster age, since the evolution during the
embedded phase has not been considered in this part of the study. Shown are the results
for star formation efficiencies of ǫ = 0.2 (blue), 0.3 (red), and 0.4 (green), as well as for a
star formation efficiency of ǫ = 1.0 (black) for comparison.
fraction of bound stars
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
time[Myr]
Figure 5.2: Shown is the fraction of bound stars as a function of the time. Here, for tsim = 0 the gas is expelled
instantaneously. Star formation efficiencies of 20% (blue), 30% (red), 40% (green), and 100% (black)
are shown. The error bars are obtained by the standard deviation.
It can be seen that the bound fraction of stars rapidly decreases within the first 20 Myr.
For a star formation efficiency of 100% the cluster is initially set up in virial equilibrium.
Such a large star formation efficiency implies the unlikely case that all gas is converted into
stars. However, a fraction of 5% of the stars is found to be ejected from the cluster in that
case. Such ejections are caused by strong few body interactions with the massive cluster
members. Here, the rather constant decrease of the fraction of bound stars indicates a
steady ejection process for at least the first 20 Myr.
The fraction of bound stars depends strongly on the star formation efficiency. For the
star formation efficiency ǫ = 0.2 the fraction of bound stars decreases quickly within the
91
5 Cluster expansion phase
first 5 Myr. Eventually, a remnant cluster of at most a few hundredth of its initial members
remains. Directly after the gas expulsion (tsim = 0) already 60% of the stars are unbound
because their velocities exceed the escape velocity of the cluster. Unbound stars leaving
the cluster lead to a further decrease of the total cluster mass and, thus, a reduction of the
escape velocity, which causes even more stars to leave the cluster. The result is a runaway
process due to a gradual decrease of the residual stellar potential by the escaping stars,
which ends in an almost total dissolution of the cluster.
For ǫ = 0.3 a bound cluster remains, although significantly reduced in stellar members.
Here, a final stellar population of ≈ 3000 stars (10%) is still bound after 20 Myr. For an
even higher star formation efficiency of ǫ = 0.4 the fraction of bound stars decreases only
to about 40%. Further results of similar investigations are summarised in Section 1.5.2.
In the following the radial evolution of the stellar members will be further detailed and
the overall importance of encounters will be investigated. Afterwards, the frequency of
encounters will be specified for the two groups of bound and unbound stars as well as their
radial dependence.
5.3 General cluster dynamics
Here, we are interested in the effect of gravitational star-disc interactions in rapidly expanding stellar populations. This has so far not been investigated since the importance of
encounters in stellar clusters has been largely underestimated in the past (see Section 1.4).
However, as shown in Chapter 4, even in sparse stellar clusters a significant number of
encounters might occur.
In accordance with Section 3.2.5, a circumstellar disc is defined as being perturbed if
> 5% of its angular momentum is removed by a stellar fly-by. As investigated in Section 4.4
the number of encounters is found to be independent of the initial disc-mass distribution
in that case. Therefore, the following results are presented for initially constant disc-mass
distributions.
Focusing on the dynamical cluster evolution after the dispersal of the gas two groups of
stars can be distinguished: (i) stars that remain as a bound population and (ii) stars, whose
velocities exceed the binding energy of the cluster. Immediately after the gas expulsion,
these two superposed groups can hardly be distinguished by observations since a detailed
determination of the stellar velocities for each spatial direction would be required. Even
if the proper motion and radial velocities were determined, the escape velocity of the
individual stars (see Equation 1.10 for a spherical, homogeneous mass distribution) remains
hard to constrain. For any unbound star that is finally separated from the remnant cluster
the cluster potential energy is further decreased so that the escape velocity strongly varies
92
5.3 General cluster dynamics
averaged number of stars per distance bin
with time. However, numerically no such limitations are given. In the following the two
groups will be treated separately while the effect of the initially overlying populations on
observations will be investigated in Section 5.4.3.
2500
total number of stars
bound after gas expulsion
bound after 10 Myr
2000
1500
1000
500
0
0.1
1
initial distance to center [pc]
Figure 5.3: Averaged number of stars per radial bin as a function of the initial distance to the cluster center for
ǫ = 0.3. Black crosses indicate the total number of stars, red circles the number of stars that are bound
to the cluster at the start of the simulation, and blue squares stars that remain bound after 10 Myr.
First, the cluster evolution is further detailed in terms of the temporal development
of the two groups. Therefore, in Figure 5.3 the number of bound stars is shown as a
function of the initial distance to the cluster center for the inner 3 pc. Comparing the total
number of stars (black line) to the number of stars that are bound instantaneously after
the gas expulsion (red line) shows that a large fraction of stars (≈ 40%, Fig. 5.2) have
high velocities that exceed their local escape velocities. The largest fraction of escapers is
initially located in the outer cluster parts (> 1 pc) since their velocities are more likely to
exceed the escape velocity. Contrary most of the innermost stars remain captured within
the inner potential well. With time the number of bound stars further decreases. After 10
Myr (blue line) more than 90% of the stars initially located in the outer cluster region are
expelled from the cluster. However, nearly half of the stars that are initially located within
the innermost cluster part (≈ 0.1 pc) remain bound, providing a peak in the number of
stars which extends up to 0.4 pc from the cluster center.
The question arises how the positions of the bound and unbound particles change with
time. In Figure 5.4 the averaged number of stars per radial bin is shown as a function
of the distance to obtain the intrinsic distribution of the stars after 10 Myr. The core
region is populated by a substantial number of bound stars that diffuse out to distances
93
averaged number of stars per distance bin
5 Cluster expansion phase
1000
100
1
10
100
distance to center after 10 Myr [pc]
Figure 5.4: Shown is the average number of stars per radial bin versus the distance to the cluster center after 10
Myr with ǫ = 0.3. Red circles indicate unbound stars, while blue squares represent bound stars. All
shown data bins contain on average more than 30 star.
of up to 20 pc. Between 10 − 30 pc a transition from mostly bound stars to an increasing
fraction of unbound stars can be determined. Outside 30 pc all stars are unbound with
the major fraction of unbound stars being expelled to distances > 50 pc. Due to the fact
that the bound and unbound fraction of stars are nearly completely separated indicates
that the enclosed mass, and therefore the escape velocity, will remain rather constant over
time. However, some stars located in the outskirts of the remnant bound population, where
bound and unbound stars still coexist, might become unbound during the following cluster
development.
In Figure 5.5 the averaged initial distance to the cluster center is shown as a function
of the distance after 10 Myr to reveal the movement of the bound and unbound stars.
As pointed out before, within 10 pc a restructured bound cluster remains. Its innermost
regions of < 6 pc mainly consist of stars that have been located within < 1 pc before the
gas expulsion process. Moreover, the vast majority of the bound stars has been initially
located within the inner 2 pc from the cluster center.
By contrast, the unbound stars are rapidly expelled from the cluster. In general, the
process of stars becoming unbound can be divided in two phases. In the first phase the
high velocity stars, which exceed the local escape velocities, are rapidly expelled from the
cluster to distances > 30 pc. Some of the stars initially located close to the cluster center
might reach large distances of up to 200 pc due to the high initial velocity dispersion in this
regime and an accompanied prominent fraction of stars residing in the high velocity tail.
94
initial distance to center [pc]
5.3 General cluster dynamics
1
0.1
1
10
100
distance to center after 10 Myr [pc]
Figure 5.5: Shown is the averaged initial distance to the cluster center as a function of the distance to the cluster
center after 10 Myr with ǫ = 0.3. Red circles indicate unbound stars, while blue squares represent bound
stars. All shown data bins contain on average more than 30 star.
Consequently, these high velocity stars have to pass the entire remaining stellar population
early on.
The loss of stars during the first phase significantly reduces the binding energy of the
cluster, which leads to a further dispersion of the population. Inferred by this second phase
preferentially stars located in the outer regions are expelled from the cluster since the local
escape velocity stabilises from inside out. In Figure 5.5 the sharp edge in the red line
around 30 pc separates such recently expelled outer stars from the major population of
unbound stars. For < 30 pc - the earlier stated transitional region where both groups of
unbound and bound stars coexist - a distinctive peak in the distribution of the unbound
population appears, with most stars initially located > 3 pc.
In total, it can be deduced that the remaining bound cluster consists of stars initially
located in 1 − 2 pc from the cluster center that are spread over several parsec after the gas
expulsion, while in particular the number of stars in the outer cluster regions is largely
reduced.
Figure 5.6 shows the temporal development of the Lagrange radii for stars that are
bound to the cluster. Within the first Myr the cluster rapidly expands due to the supervirial velocities of the stars. For stars that are initially located inside 1 pc distance to
the cluster center the radial distance increases by about one magnitude. This means that
stars initially located within 0.1 pc reach a virial equilibrium state for 1 pc distance from
the cluster center after about 1 to 2 Myr. It implies that after such short time scales
95
5 Cluster expansion phase
distance to cluster center [pc]
1%
5%
20%
30%
50%
70%
90%
10
1
0.1
0.1
1
10
time [Myr]
Figure 5.6: Lagrange radii as a function of the simulation time for the bound fraction of stars. The Lagrangian radii
are presented for enclosed mass fractions of 1%, 5%, 20%, 30%, 50%, 70%, and 90%.
all unbound stars have been expelled from within this inner region. With time the cluster
further expands and the unbound stars separate from the remnant bound population. After
about 10 Myr the innermost 30% of the bound stellar mass are found virialised and the
vast majority of the bound population is located within 20 pc from the cluster center.
Having investigated the motion of the stars, another important quantity of the stellar
cluster is its local volume density, which is shown in Figure 5.7 for the bound fraction of
stars as a function of the time. After the expulsion of the residual gas the local density
drops rapidly below 104 stars per pc−3 . It can be seen that stars that have been part of
a disc-perturbing encounter event (red line) are initially located in the dense parts of the
cluster. By contrast, stars with unperturbed discs (black line) reside in regions, which are on
average half as dense. After a few Myr the expansion of the cluster population decelerates
since part of the high kinetic energy of the stars is converted into potential cluster energy.
Stars with perturbed discs approach a new state of equilibrium after about 2 Myr while
for stars with unperturbed discs the curve flattens slightly delayed. Considering that the
major fraction of bound stars with perturbed discs remains in the innermost regions of the
cluster this is in accordance with our previous results (Sec. 1.4.2).
5.4 Importance of encounters
A first step to qualify the importance of encounters during the expansion phase of the cluster
is to investigate the actual number of perturbed discs as a function of the simulation time,
which has been plotted in Figure 5.8. The results are presented for the total number of
96
averaged local volume density [pc−3 ]
5.4 Importance of encounters
stars with perturbed discs
stars with unperturbed discs
100000
10000
1000
100
0.1
1
10
tsim [Myr]
Figure 5.7: Shown is the averaged local number volume density as a function of the simulation time for the bound
stellar population. The red line indicates stars that were part of a disc-perturbing encounter event and
the black line represents stars with unperturbed discs.
stars (Ntotal,pert /Ntotal , dashed line) as well as the bound (Nbound,pert /Nbound , solid line)
and unbound (Nunbound,pert /Nunbound , dotted line) fraction. It can be seen that the total
fraction of perturbed discs increases rapidly within the first 105 yr, since the cluster density
remains sufficiently high, but that the curve flattens subsequently to a total of around 10%
of the discs being perturbed. During the further expansion of the cluster the total fraction
of perturbed discs remains constant since the stellar density drops significantly (Fig. 5.7).
Focusing on the large fraction of unbound stars a similar picture is obtained. After a
rapid increase shortly after the expulsion of the gas the fraction of perturbed discs remains
constant at 9%.
For the fraction of stars that remain bound as a cluster, we find again a similar amount
of around 10% of the bound stars being part of a recent disc-perturbing encounter event
within the first 105 yr, where the cluster is still very compact. However, during the further
expansion of the cluster the fraction of perturbed discs increases significantly to > 25% until
the cluster reaches its new equilibrium state. This might either indicate that encounters
of the bound residual cluster members after the gas expulsion phase are a frequent event
or that an increased fraction of stars with unperturbed discs becomes unbound. Although
the total number of stars has been found to remain rather constant the former has to be
considered due to the much lower number of bound stars.
To frame the prominent process the sub-group of stars that remain bound after 20 Myr
has been plotted in Figure 5.9. It is found that the majority of discs surrounding finally
bound stars (24%) are perturbed early on. However, a rather small amount of 2% is part
97
5 Cluster expansion phase
x = bound
unbound
total
Nx,perturbed /Nx
0.25
0.2
0.15
0.1
0.05
0
5
10
15
20
tsim [Myr]
Figure 5.8: Shown is the fraction of stars with perturbed discs as a function of the time for the sub-samples of bound
(solid line), unbound (dotted line), and total number of stars (dashed line). The filled curves indicate
the standard deviation. For the total fraction the error is below 0.5%.
0.28
0.26
Nb,perturbed /Nb
0.24
0.22
0.2
0.18
0.16
0.14
0.12
finally bound subgroup
0.1
0
5
10
15
20
tsim [Myr]
Figure 5.9: Like Figure 5.8 but focusing only on the sub-group of stars that are bound after 20 Myr.
98
5.4 Importance of encounters
of an encounter in the subsequent developmental phase as can be deduced from the slightly
increasing fraction of perturbed discs.
Hence, supported by the results for the dynamical cluster evolution, here, it can be
concluded that the significantly differing fraction of perturbed discs, as seen in Figure 5.9,
is caused by preferentially stars in the cluster outskirts becoming unbound.
5.4.1 Dependence on stellar mass
all stars
Number of encounters per star
2
high-mass stars
1.5
1
0.5
0
0
5
10
15
20
tsim [Myr]
Figure 5.10: The number of encounters per star as a function of the time. Here, the focus is on the sub-samples of
the high-mass stars (mstar > 7 M⊙ , black line) and the total number of stars that are bound after 20
Myr (red line).
Investigating stellar interactions in sparse environments, the fraction of high-mass stars
is of unique importance (see also Section 4.4.2). Figure 5.10 shows the average number of
encounters per star as a function of the time after gas expulsion for the total number of
stars and explicitly for the number of high-mass stars (mstar > 7 M⊙ , which equate to a
number fraction of 1% of the total number of stars). As before, the focus here is on the
sub-group of finally bound stars. From the total number of bound stars can be deduced
that multiple encounter events are very rare with on average one out of four stars being part
of an encounter, which is in accordance with the previously determined absolute fraction
of perturbed discs (solid line in Fig. 5.8).
However, the number of encounters per high-mass star deviates considerably from these
results. Within the first 105 yr on average a smaller fraction of massive stars are found
being part of a disc-perturbing encounter, which can simply be explained by the much
weaker perturbations of their discs due to the on average lower encounter mass ratios.
99
5 Cluster expansion phase
With time the number of encounters increases significantly so that after 20 Myr the bound
high-mass stars are on average part of around two encounter events. One explanation
might be that significantly more massive stars remain bound. However, we compared the
fraction of escaping high-mass stars to the total fraction of unbound stars as well as the
half-mass radii of these two groups and determined no differences between the two mass
groups. This is not surprising since the stellar velocities are scaled independent of the
stellar mass and dynamical mass segregation can be excluded on such short time scales.
Consequently, the apparent increase in encounters per high-mass stars can be retraced as
gravitational focusing.
It can be concluded, that massive stars might be part of an encounter event even after
the dispersion of the remaining population due to gravitational focusing while the vast
majority of all encounters occurs at the start of the expansion phase.
5.4.2 Evidence for disc destruction
bound subgroup
unbound
fraction of destroyed discs
2%
1%
0%
0
5
10
15
20
tsim [Myr]
Figure 5.11: Shown is the fraction of destroyed discs as a function of the time after the gas expulsion. The fraction
of unbound stars is shown as a dotted line and the sub-group of stars that remain bound after 20 Myr
is shown in solid line. The filled curves indicate the standard deviation.
So far the focus was on gravitational perturbations of the circumstellar discs in expanding
cluster environments. However, the resolution of today’s observational instruments is not
precise enough to dissolve such weak interaction features. As discussed in Section 1.3 a
fundamental quantity observed is the cluster disc fraction. In Figure 5.11 the fraction discs
that are destroyed after the gas expulsion is shown as a function of the time. Besides the
group of finally unbound stars (dotted line) the sub-group of finally bound stars is presented.
100
5.4 Importance of encounters
For the latter, the bound stars after 20 Myr were determined and their evolution was traced
back in time.
While only a minor fraction of < 1% of the unbound stars loose their disc in an encounter
we find an increasing fraction of 1 − 2% for the bound stars. Although the total fraction
of destroyed discs remains rather low a clearly diverse evolution can be determined for the
fraction of bound and unbound stars. Again, the bound stars being more frequently located
in the dense inner cluster regions leads to strong encounter events early on and even some
encounters during the further development as indicated by the slightly increasing number
of perturbed discs.
This picture is completely opposite to what is found for virialised clusters, where stars
preferentially become unbound by ejections, loosing their discs most likely in the appendant
encounter event (Sec. 4.3).
5.4.3 Dependence on distance to cluster center
Observations reveal that stellar clusters might expand to rcluster > 10 pc within the first 10
Myr of their lifetime (Fig. 1.8). While the definition of the size of a cluster appears very
straightforward in case of numerical simulations, since the bound and the unbound stars
can be simply separated by calculating the kinetic and potential energies of the cluster
members, the definition of a cluster size (or even of a cluster itself) for comparisons to
observations is very challenging in the first Myrs since bound and unbound stars can be
rarely distinguished.
Observed cluster sizes are usually estimated by the half-mass radius or the median radius
of a certain stellar mass range, like done by Wolff et al. (2007) for B-type stars. In general,
a limiting surface density threshold to the surrounding field stars (Borissova et al., 2008) or
alternatively a fixed maximum radius (Goodwin & Bastian, 2006) is required to constrain
the considered cluster area (see also Pfalzner & Kaczmarek, 2013). However, observations
of circumstellar discs are in most cases restricted to the inner cluster regions of < 5 pc
(e.g. Lada et al., 1993; Hillenbrand et al., 1998; Haisch et al., 2001a; Sung et al., 2009) due
to the high number of stellar candidates but depend considerably on the observed cluster
and used instruments. Due to the lack of a beneficial parameter range for comparisons to
observations, here, the focus will be on the radial dependence of the fraction of perturbed
discs.
Figure 5.12 shows the influence of the considered cluster region on the total fraction of
perturbed discs for different time steps. The fraction of perturbed discs is shown in red and
the cumulative number of stars is shown in blue. 0.1 Myr after the gas expulsion (Fig. 5.12a)
a strong radial dependence of the fraction of perturbed discs can be determined with more
than 30% perturbed discs in the cluster core region (rcore = 0.3 pc). If the entire cluster
101
5 Cluster expansion phase
a)
0.8
after 0.1 Myr
0.6
Ntotal (r)/Ninitial
Nperturbed (r)/Ntotal (r)
0.7
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
radius [pc]
b)
0.8
after 6.0 Myr
0.6
Ntotal (r)/Ninitial
Nperturbed (r)/Ntotal (r)
0.7
0.5
0.4
0.3
0.2
0.1
0
2
4
6
8
10
12
14
radius [pc]
Figure 5.12: The fraction of perturbed discs within a sphere of radius r (red line, left y-label) and the cumulative
number of stars (blue line, right y-label) as a function of the distance to the cluster center within the
first (a) 0.1 Myr and (b) 6 Myr after the gas expulsion.
102
5.5 Discussion
is considered their fraction drops down to only 10%. Hence, as expected, the majority of
stellar interactions takes place in the dense central region.
With time the unbound stars are expelled from the cluster and the remnant bound
population expands significantly so that after 6 Myr (Fig. 5.12b) less than 15% of the
total initial population remain located within 15 pc distance to the cluster center, while
the half-mass radius for the bound population is found around 10 pc. However, more than
300 stars (∼ 10% of the remnant cluster) remain bound within the inner 2 pc. Although
it takes several Myr for the cluster to completely separate the bound from the unbound
stars, the re-virialisation process is completed at first in the inner cluster parts while the
cluster outskirts might still contain a few escapers.
Focusing on the inner 1 pc of the remnant cluster, half of the discs are found disrupted,
while a perturbed fraction of around 20% is obtained if all stars within the full cluster
size of 15 pc would be considered. The rare cases of encounters after the gas expulsion
preferentially take place in the dense central region of the cluster preserving an increase
of the total fraction of perturbed discs towards the cluster center. Due to the cluster
expansion these stars have most likely been initially located even closer to the cluster
center where encounter events were prominent, e.g. stars within 1 pc from the center are
initially found within an average distance of 0.2 pc (Fig. 5.5). The spread of the stellar
population consequently leads to an outward increasing fraction of perturbed discs with
time.
5.5 Discussion
Prior to the here studied gas expulsion phase the stars were exposed to the cluster dynamics
in the embedded phase. However, in this chapter, all stars deliberately started out with
an unperturbed disc. The benefit of this approach is that the effect of encounters during
the expansion process can be separated from the prior development during the embedded
phase. Considering the early interaction history before the gas expulsion would naturally
provide larger values for the fraction of perturbed and destroyed discs than obtained here.
The prior evolution in a virial cluster has been investigated in the previous chapter and its
impact on the overall disc losses will be discussed in Chapter 6.
Stellar evolution has been neglected since it only becomes significant on much longer
time scales than investigated here. The first supernovae explosions of the most massive
stars are expected after a few Myr. However, the fraction of massive stars compared to the
remnant population is very low, so that the mass reduction would be of the order of 5%
during the first 10 Myr. The mass reduction is expected to proceed steadily so that the
induced cluster expansion would be nearly adiabatic and, thus, insignificant. Nevertheless,
103
5 Cluster expansion phase
stellar evolution eventually leads to a further dispersion of the remnant cluster and an
increasing fraction of outer stars becoming unbound. This implies that the present results
underestimate the expansion process on larger time scales (> 10 Myr).
Another simplification is the neglect of primordial binaries. This will additionally underestimate the expansion process since the cluster dynamics are generally influenced by fewbody interactions (Hills, 1975; Heggie, 1975). Again, the consequence would be a further
dispersion of the stellar population, which would increase the here investigated expansion
effect. However, the released energy would be small since few-body interactions become
increasingly rare in the expanding clusters. Pfalzner & Kaczmarek (2013) determined a
total fraction of about 5% of the stellar members being ejected from the clusters due to
stellar interactions without considering primordial binaries. Including primordial binaries
would presumably increase this fraction. Future investigations are needed to constrain the
importance of binaries.
In this work, the focus was on the effect of the cluster expansion on encounter-induced
disc losses. However, the rapid drop of the local density implies that not only encounters
but as well other external disc destruction processes like photo-evaporation due to massive
stars (e.g. Scally & Clarke, 2001; Clarke et al., 2001; Johnstone et al., 2004; Drake et al.,
2009) become less important for expanding clusters. Here, the reduced destruction rates are
induced by the increasing radial distances to the O-stars (Johnstone et al., 1998; Scally &
Clarke, 2001, and references therein). So the question arises how (or if at all) circumstellar
discs further disperse after the expulsion of the gas.
5.6 Summary
Not only in the embedded phase but as well in the gas expulsion phase circumstellar disc
properties are influenced by stellar encounters. After the gas expulsion, which usually
occurs in less than a crossing time for the here investigated massive clusters, only a small
fraction of stars remains bound as a cluster, whereas the larger part of the stellar population
is expelled. The super-virial cluster rapidly expands while preferentially the outer stars with
unperturbed discs become unbound. The main results are the following:
1. The average distance of the remnant bound stars to the cluster center increases rapidly
by about a factor of ten during the first 5 Myr after the gas expulsion. Consequently,
stellar interactions are only prominent during the first few 105 yr after the gas expulsion. Afterwards, the local stellar densities drop rapidly below 104 stars per pc−3
where encounters are rare.
2. 25% of the stellar members that remain bound experience an encounter during the
104
5.6 Summary
cluster expansion phase. By contrast, only 10% of the stars that become unbound
are part of an encounter event in this phase.
3. As preferentially stars from the sparsely populated cluster outskirts become unbound
the field star population has a higher disc frequency than those typical observed in
clusters of age > 5 Myr.
4. As an artifact of cluster expansion the observed disc frequency seems to drop. Consequently, the cluster disc fractions might provide strong upper limits for the survivability of circumstellar discs.
For the most massive clusters in the solar neighbourhood it takes ∼ 3 Myr for the stars
that initially populated the dense cluster core region (rcore = 0.3 pc) to fill the inner 3 pc
of the remnant cluster. As the disc fraction increases with distance to the cluster center
the cluster expansion mimics a non-existent decrease in the disc fraction with cluster age.
In how far this influences the results of the disc destruction as a function of cluster age
requires further investigation as here not only external processes but as well internal disc
destruction processes, like viscous torques, turbulent effects, and magnetic fields, play a
role.
Nevertheless, early external destruction processes in the dense cluster core region might
have to a large degree indirectly shaped observational results for the lifetime of circumstellar
discs.
105
5 Cluster expansion phase
106
6 Conclusion
There is strong observational evidence that in stellar clusters the protoplanetary discs surrounding new born stars disappear within ∼ 6 Myr after their formation (Haisch et al.,
2001b; Hillenbrand, 2002; Sicilia-Aguilar et al., 2006; Hernández et al., 2007; Currie et al.,
2008; Hernández et al., 2008; Mamajek, 2009; Massi et al., 2010). One of the key questions in this context is: Which physical mechanism dominates the rapid evolutionary disc
destruction processes?
Among the established processes are internal processes such as viscous torques (e.g. Shu
et al., 1987), turbulent effects (Klahr & Bodenheimer, 2003), and magnetic fields (Balbus
& Hawley, 2002), as well as external disc destruction processes like photoevaporation and
winds from the massive stars (e.g. Scally & Clarke, 2001; Johnstone et al., 2004; Ercolano
et al., 2008), as well as the here investigated gravitational star-disc interactions (e.g. Hall
et al., 1996; Pfalzner, 2004; Moeckel & Bally, 2006). However, an effect that has so far
been neglected when comparing disc fractions in different star clusters is their rapid expansion after the gas expulsion. This expansion significantly influences the interpretation
of observed cluster disc frequencies on time scales > 3 Myr.
In this study, the focus has been on the most massive clusters in the solar neighbourhood, since for these clusters the entire expansion process can be followed by observations
(Pfalzner, 2009; Portegies Zwart et al., 2010), whereas clusters of lower mass probably dissolve after the expulsion of their natal gas. In the light of our results one can conclude
that the disc destruction process can be separated in an early phase, where the influence of
the external processes plays a significant role, as well as a phase after the expulsion of the
remnant gas, where the cluster expands and internal destruction processes might dominate.
Here, the losses during the two separated evolutionary phases, namely the embedded and
the expanding cluster phase, have been quantified by investigating the effect of the initial
disc-mass distribution in star-disc encounters.
It remains unclear, which process dominates during the early embedded destruction
phase. An often favoured destruction mechanism is photoevaporation by the surrounding
massive stars, which according to numerical simulations might destroy some of the discs
in the near vicinity of the high-mass stars on time scales as short as < 105 yr (Scally &
Clarke, 2001; Gorti & Hollenbach, 2009). By contrast, recent observations of evaporating
6 Conclusion
discs in young star clusters found that even in strong radiation fields the circumstellar
material might remain bound much longer than previously assumed (Vicente et al., 2013).
Disc destruction by photoevaporation of the surrounding massive stars is prominent in the
outer disc parts (Balog et al., 2008). Therefore, assuming an initially steep distribution of
the disc material would suggest a minor effect of photoevaporation on the relative mass
losses of circumstellar discs, while the effect would be potentially larger if an initially flat
disc-mass distributions is assumed.
Stellar encounters, although often discarded as probably unimportant, are another significant destruction process during the early embedded phase (e.g. Pfalzner et al., 2005b;
Moeckel & Bally, 2006; Olczak et al., 2010). In contrast to what would be expected for
the disc destruction by photoevaporation, in this study it has been found that the effect of
encounters on the cluster disc fraction is independent of the initial disc-mass distribution
(Chapter 4), with a large fraction of stars losing their discs completely. The highest influence on the disc losses has been found in dense cluster environments, where interactions
between the low-mass stars become increasingly important. In such dense environments the
time scales for individual disc destruction can be considerably short (∼ 104 yr, Eq. 1.12)
and stellar interactions are frequent.
After the gas expulsion the local stellar density drops considerably. Consequently, in the
later disc destruction phase external processes become less important and internal processes
might dominate. The remnant gas is usually expelled quickly from the cluster (< 5 Myr
Whitworth, 1979; Leisawitz et al., 1989; Lada & Lada, 2003). For the here investigated
massive clusters this time scale might be even shorter due to a significantly high fraction
of massive stars that are primarily responsible for blowing out the residual gas, thus, an
expulsion of the remnant gas within the first 1 Myr seems reasonable. The gas expulsion
leads to a spreading of the core regions of massive clusters from 0.3 pc to about 3 pc in 2 − 3
Myr, which translates into a cluster age of 3 − 4 Myr if we assume an age of 1 Myr at the
end of the gas expulsion phase.
A comparison of the present results to observations has to be done carefully, since our
results are limited to massive clusters and, thus, provide a lower limit for the time scales
of the expansion effect. The star clusters, which shape the observed cluster disc frequency,
are typically not as massive (< 104 M⊙ , e.g. Haisch et al., 2001b) as the here investigated
populations. In such less massive clusters the number of high-mass stars is reduced and
the gas expulsion process lasts longer (∼ 3 Myr), so that the effect might become apparent
after ∼ 5 Myr.
The observations of the circumstellar disc frequencies are generally limited to the inner
parts of the stellar clusters. Hereby, the typical sizes and the number of stars considered in
the surveys vary between 0.3 to 3 pc and a few to several hundredths of stars, respectively
108
(Haisch et al., 2001b; Hernández et al., 2008; Mamajek, 2009). The dimensions of the
observed regions are in most cases simply determined by the beam size of the telescopes
and the distances to the observed populations. A consequence is that generally more
stars are observed for young compact clusters (> 100), while for the sparse, expanding
populations the fraction of observed stars is significantly reduced (< 50) (see e.g. Haisch
et al., 2001b). This might naturally result in a decreasing disc fraction - simply explained
by the expansion of the dense cluster core.
Two typical clusters that are massive and old enough to have expelled their gas and
reached the leaky cluster sequence are NGC 2244 and NGC 6611. Balog et al. (2007)
analysed the spatial distribution of circumstellar discs in the 3 − 4 Myr-old cluster NGC
2244, which is situated in the young part of the leaky cluster sequence. They found a
cluster disc fraction of 44.5% for the innermost 2.5 pc of the cluster. Oliveira et al. (2005)
performed a similar study for NGC 6611 and found a cluster disc fraction of 58% in the
inner 3 pc of the cluster. As these clusters already expelled their gas, the disc frequencies
determined for these clusters are fundamentally altered by the cluster expansion process.
The core of the clusters most likely has been much more compact (> 104 pc−3 ) before the gas
expulsion, so that densities have been large enough for external processes to substantially
shape the currently visible disc frequency.
Another consequence of the gas expulsion is that the large fraction of stars, that are
expelled into the field, remain surrounded by a circumstellar discs. This suggests a higher
probability of planetary systems around stars in the field than in stellar clusters. In this
context, Gilliland et al. (2000) performed extensive observations of 34 000 stars in the
13 Gyr old cluster 47 Tucanae but found no evidence for planetary systems, which was
contrary to what has been expected according to its present stellar density. Despite the
general explanations like crowding or a low metallicity of the cluster (Weldrake et al., 2005)
the present results emphasise that these clusters have been much more compact with most
of today observed stars having formed much closer to the cluster center so that the cluster
expansion itself provides an additional, non-negligible effect for the absence of planetary
systems and promotes an early on destruction of the discs.
Concludingly, the present results provide valuable information for the further interpretation of observational data from upcoming telescopes. The Atacama Large Millimeter Array
in Chile, which just recently started operating, and the James Webb Space Telescope to
be launched in 2018 will both provide unprecedented high-resolution data that will for the
first time allow a detailed mapping of protoplanetary discs. In particular, a qualitatively
and quantitatively enhancement of observational samples of circumstellar disc in massive
clusters might provide new insights in the dynamical cluster evolution.
109
6 Conclusion
110
7 Summary
In this study the relevance of star-disc encounters in massive star clusters during the crucial
first 15 Myr after their formation has been investigated. The emphasise of the investigation
was on the influence of the initial disc-mass distribution and the cluster expansion phase
after the gas expulsion process. The most relevant results are the following:
1. Performing a parameter study of individual star-disc encounters, significant differences between the investigated disc-mass distributions were only found for interactions
with high-mass stars that mainly perturbed the outer parts of the disc. In addition it
was found that independent of the initial distribution encounters may produce steep
density profiles of p > 2 as possibly required for the formation of planetary systems
like our own Solar System.
2. Nevertheless, the simulations of the early embedded phase demonstrated that the
fraction of disc-less stars is largely unaffected by the initial disc-mass distribution.
Only for very high cluster densities differences become apparent with 60% of discs
being destroyed in case of initially constant disc-mass distributions while for initially
steep disc-mass distributions (Σ ∝ r−7/4 ) only 40% are destroyed. The reason is that
in very dense clusters stellar interactions interactions of low-mass stars dominate,
which show the largest dependency on the initial disc-mass distribution due to the
generally high encounter mass ratios.
3. After the expulsion of the residual gas the stellar density drops rapidly and the initial
core region of the cluster spreads by about a factor of ten. As a result, star-disc
encounters are only prominent during the first few 105 yr after gas expulsion. In
this phase the initial disc-mass distribution is of minor importance for the results,
since usually no discs are destroyed and the total number of encounters is sufficiently
low. Preferentially stars from the outer cluster regions become unbound during the
expulsion process and are, thus, to a much lower degree affected by encounters.
4. The rapid expansion of the star clusters mimics a drop in the observed cluster disc
frequency, which has so far been largely neglected when comparing disc fractions
in different clusters. Furthermore, observations of the cluster disc fraction in populations older than 3 Myr likely provide strong upper limits for the survivability of
7 Summary
circumstellar discs since the large fraction of disc-surrounded stars is expelled from
the cluster. Consequently, the formation of planetary systems is benefited around
stars in the field population.
112
A Analytical investigations of star formation
efficiencies
Hills (1980) analysed the evolution of the total cluster energy during the gas expulsion
phase. If the system is assumed to be in a virialised state before gas expulsion the virial
theorem is given by 2T0 + W0 = 0, with initial kinetic energy T0 = 21 Mtot σ02 and initial
2 /2R . Here, M
potential energy W0 = GMtot
0
tot = Mstars + Mgas represents the initial total
mass of the cluster and R0 the effective cluster radius. If a system of equal mass stars is
assumed, R0 would be obtained by the mean distance between each pair of stars in the
cluster. The velocity dispersion prior to the gas mass loss is obtained as
σ02 = GMtot /2R0 .
(A.1)
The total energy instantaneously after gas expulsion is given by
2 2 2
2
1
1
Gǫ Mtot
GM 2
=
,
E =T +W =
M σ0 −
ǫMtot σ0 −
2
R0
2
R0
(A.2)
where ǫ = M/(Mstars + Mgas ) = M/Mtot is the star formation efficiency of the cluster, M
defines the cluster mass after the gas is expelled. If an adiabatic gas expulsion process,
on timescales larger than the crossing time, is assumed, the cluster will remain in a virial
equilibrium state with the stars adapting to the reduced gravitational potential. However,
if the gas is expelled instantaneously the velocity dispersion will be similar to the primordial
dispersion before gas expulsion.
If the new state of equilibrium is reached, the total energy is given by
GM 2
.,
(A.3)
4R
where R is the new equilibrium cluster radius. Combining Equation A.1, A.2, and A.3
results in
E = −W/2 = −
1
R
=
R0
ǫ
in case of a gradual gas mass loss while rapid gas expulsion leads to
(A.4)
A Analytical investigations of star formation efficiencies
ǫ
R
=
.
R0
2ǫ − 1
(A.5)
It can easily be deduced that in case of an impulsive mass loss a star formation efficiency
of ǫ > 0.5 is required for a subsequently bound cluster (see e.g. Hills, 1980, for further
details).
Boily & Kroupa (2003a) reviewed the analytical approach by Hills (1980) and showed
that the fraction of bound stars is not only a function of the star formation efficiency but
also of the initial stellar distribution function F (r, v). The fraction of bound stellar mass
Nbound /Ntotal at radius r is defined as
R ve
R ve
3
2
f (v) v 2 dv
Nbound
ve,⋆ F (r, v) d v v dv
v
R ve,⋆
= R ve
−
1
=
− 1,
e
3
2
2
Ntotal
0 F (r, v) d v v dv
0 f (v) v dv
(A.6)
where ve and ve,⋆ are the local escape velocities before and after the gas expulsion, respectively. f (v) is the stellar velocity distribution function defined by the stellar distribution
function at constant radius r. Assuming a power-law velocity distribution function of
1.0
0.9
β = ...
N(bound) / N(initial)
0.8
0.7
0.6
0.5
−3/4
−1/2
0
0.4
+1/2
+3/4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
s.f.e.
Figure A.1: Relative fraction of bound stars as a function of the star formation efficiency for power-law velocity
distribution functions with exponent β − 2. The figure is taken from Boily & Kroupa (2003a).
f (v) d3 v ∝ v β−2 d3 v ∝ v β dv
114
(A.7)
the fraction of bound stars can now be calculated by using ve,⋆ (r) = ǫ1/2 ve (r) and Equation A.6. For j iterations with j → ∞ the fraction of bound stars is given by
1+β
Nbound
= ǫ 1−β .
Ntotal
(A.8)
The dependence of the bound stellar fraction on the exponent β of the stellar velocity
distribution is shown in Figure A.1. It can be seen, that the fraction of bound stars
decreases rapidly if the velocity distribution is prone to high-velocity stars with β > 0.
In case of a Plummer velocity distribution function Boily & Kroupa (2003a) find a bound
fraction of 86% for ǫ = 0.5. However, although a crucial improvement to the investigations
of Hills (1980) has been found, numerical simulations are indispensable to obtain proper
rates of bound stars (Boily & Kroupa, 2003a,b).
115
A Analytical investigations of star formation efficiencies
116
B Numerics of individual star-disc encounters
The Runge-Kutta Cash-Karp Integrator
Since only low-mass discs are investigated in this work, gravitational interactions between
the disc particles are neglected, which reduces the simulation effort to 2N + 1 calculations
(between the N disc particles and each star as well as between the two stars). In the single
star-disc encounter simulations the ordinary differential equations with given initial conditions y(t0 ) = y0 have been solved via a Runge-Kutta Cash-Karp integrator as described
in Press et al. (1992). This integrator is based on the Runge-Kutta method but has been
extended by an adaptive time step size control.
In contrast, the Runge-Kutta method propagates a solution over the interval [tn , tn +
∆t] by combining the information from several Euler method steps and then using the
information obtained to match a Taylor series expansion up to some higher order. So the
approximation for yn+1 is given by a weighted average of approximated values of fk at
several time steps within the interval [tn , tn + ∆t] with intermediate values
ki = f (tn + ci ∆t, yn +
i−1
X
j=1
aij kj ) · ∆t
with i = 1, ..., s.
(B.1)
The next coordinate of the phase space can now be interpolated by
yn+1 = yn +
s
X
bi k i .
(B.2)
i=1
To specify this particular method, one needs to provide the number of stages s, and the
coefficients aij (for 1 ≤ j < i ≤ s), bi (for i = 1, 2, ..., s) and ci (for i = 2, 3, ..., s). These
data are usually arranged in a mnemonic device, known as a Butcher tableau (see Fig. B.1).
B Numerics of individual star-disc encounters
0
c2 a21
c3 a31 a32
..
..
.
.
cs
..
.
as1 as2 · · · as,s1
b1
b2
· · · bs−1 bs
Figure B.1: Structure of a Butcher tableau.
For example, the fourth-order Runge-Kutta method will lead to
k1 = ∆tf (tn , yn )
k1
∆t
, yn + )
2
2
∆t
k2
k3 = ∆tf (tn +
, yn + )
2
2
k4 = ∆tf (tn + ∆t, yn + k3 )
k2 = ∆tf (tn +
⇒ yn+1 = yn +
k1 k2 k3 k4
+
+
+
+ O(h5 ).
6
3
3
6
where ki with i = 1, 2, 3, 4 are the intermediate steps and yn+1 is the approximation for
time step tn+1 .
This integrator is much more expensive in computation time than the Euler method, yet
leads to much smaller errors and therefore to stable orbits of the particles.
Adaptive time step control
As already mentioned in Section 2.1.1 a time step control can significantly enhance the
computational effort of an inhomogeneous system and at the same time maintain a certain
accuracy of the simulation code. A high order integrator, using an adaptive time step
scheme, is given by the Runge-Kutta Cash-Karp integrator (RKCK ), which combines a
fifth- and fourth-order method to obtain an error estimate. Moreover, it uses six function
evaluations to calculate fourth- and fifth-order accurate solutions. The difference between
these solutions provides the error estimate of the fourth order solution which is very convenient for adaptive step size integration algorithms. If the error exceeds a specific value
the integration can be repeated with a smaller time step, which is derived from the error
estimate.
The corresponding Butcher tableau is shown in Fig. B.2 where the first row of b coefficients (see configuration of a Butcher tableau in Fig. B.1) gives the fifth-order accurate
118
0
1
5
3
10
3
5
1
7
8
1
5
3
40
3
10
11
− 54
1631
55296
37
378
2825
27648
9
40
9
− 10
5
2
175
512
0
0
6
5
70
− 27
575
13824
250
621
18575
48384
35
27
44275
110592
125
594
13525
55296
253
4096
0
277
14336
512
1771
1
4
Figure B.2: Butcher tableau for a RKCK integrator.
solution zn+1 , and the second row indicates the fourth-order solution yn+1 . The error
estimate is given by
∆i = zn+1 − yn+1 =
6
X
i=1
(bzi − byi )ki ,
(B.3)
where ∆ scales as (∆t)5 . If a step size ∆t1 is taken, which produces an error of ∆1 , the
step size ∆t0 , that would produce another error ∆0 , can be estimated as
1/5
∆0 ∆t0 = ∆t1 .
∆1
(B.4)
The desired accuracy can now be denoted by ∆0 . On the one hand, if ∆1 is larger than
∆0 in magnitude, the equation tells how much to decrease the step size for retrying the
present step, on the other hand, if ∆1 is smaller than ∆0 , the equation tells how much the
step size can be safely increased for the next step. In particular, the accuracy is adjusted to
the strength of the particle interaction, which is in the performed simulations given by the
nearness of the particles relative to the stars, e.g. close particles require a smaller global
time step. Here, the minimum step size was set to ∆tmin = 0.1 tsim and the maximum to
∆tmax = 100 tsim .
One has to note that a reduction of the time step size does not always increase the
accuracy because it increases simultaneously the accumulated (round-off) error due to the
larger number of integration steps.
The tree method
For the single encounter simulations the method used to adopt the physical problem to the
spatial domain is based on the hierarchical tree algorithm originally developed by Barnes
& Hut (1986). The technique involves a tree-like hierarchical subdivision of the simulation
119
B Numerics of individual star-disc encounters
space into cubic cells, each of which is recursively divided into 23 sub-cells or twigs whenever
more than one particle is found to occupy the same cell. The process starts from a root
cell, which contains all particles, and is successively subdividing the cells until each particle
has its own cube and therefore is representing a leaf. For each sub-cell the total mass, the
center of mass coordinates, and even higher multipole moments are computed. Since here
only gravitational forces are considered odd order multipole moments vanish.
Figure B.3: Example of the sub-division of the simulation space (left) and corresponding tree-like structure (right)
of Barnes & Hut (1986) with eight bodies.
In the force evaluation step only particles from nearby cells of a particle need to be
treated directly, while particles in distant cells are grouped to pseudo-particles located at
their center of mass. This can dramatically reduce the number of required force calculations.
The force of the pseudo-particles is usually expanded into multipoles so that the accuracy of
the algorithm can be controlled via the order of expansion. Other advantages of this method
are the freedom from geometrical assumptions and restrictions as well as the applicability
to a wide class of systems.
The force on each particle is computed by traversing the tree from the root. The spatial
group size rgroup of particles that are confined to a single pseudo-particle is compared to
the spatial distance rdist between the current particle and the pseudo-particle. This occurs
via the Multipole Acceptance Criterion (MAC),
rgroup
< rdist ,
θ
(B.5)
that is controlled by the accuracy parameter θ. If the current particle is more distant than
rgroup /θ, the cell is used to compute the force on the chosen particle. If not, then the
algorithm is recursively applied to the cells on the next level (children). The choice of θ
determines a compromise between accuracy (θ ≪ 1) and computational speed (θ ≥ 1). The
recommended range of values for the threshold parameter is given by 0.1 < θ < 1.0. Larger
120
values will result in low accuracy whereas for a much lower value the scheme will approach
the O(N 2 ) limit.
In case of a homogeneous distribution the computational effort of this method scales as
O(N log N ), because n divisions of a volume V with N particles are required to place one
particle per box of average volume V /N ,
V
V
log N
∼ n ⇒ 8n ∼ N ⇒ n ∼
∼ log N .
N
8
log 8
(B.6)
In this work the Treecode is not used to investigate particle-particle interactions, because
these can be neglected due to the low-mass of the concerned discs, but the effects of the
disc particles on the stars are calculated by this method as they are altering the orbit of the
stars. For more details of the implementation of the code see Pfalzner & Gibbon (1996).
121
B Numerics of individual star-disc encounters
122
C Stellar number density profiles
The choice of an initial density profile after the gas expulsion is not straight forward due to
the rapid evolution and limited observations. An isothermal density distribution function as
used for the embedded clusters seems misrepresenting since encounters in the early lifetime
of the clusters significantly change the profiles within these first Myr of the embedded phase,
as demonstrated in Section 4.2. The mostly used distribution functions for observations
and numerical models are described in the following.
Plummer model
Plummer (1911) showed that stellar density profiles of observed globular clusters can be
fitted by a potential, today known as Plummer potential, which is given by
GMcluster
φ(r) = −
a
r2
1+ 2
a
−1/2
(C.1)
with gravitational constant G, total mass of the cluster Mcluster , and the Plummer scale
length a, which defines the size of the inner core. Furthermore, a is related to the half-mass
radius by rrm ≈ 1.305a. The according Plummer density distribution is given by
3Mcluster
ρPlummer (r) =
4πa3
r2
1+ 2
a
−5/2
.
(C.2)
Such Plummer potentials represent globular clusters that are up to several Gyrs (!) old,
they are not necessarily a good description for younger clusters. Nevertheless, it is one of
the most common used stellar cluster models to sample stellar distributions of any stellar
age because of its simplicity (e.g. Aarseth et al., 1974; Kroupa et al., 2001; Goodwin &
Bastian, 2006; Bastian & Goodwin, 2006; Baumgardt & Kroupa, 2007; Allison et al., 2009;
Brasser et al., 2012).
King model
Another often used model is the set of so-called King models. They were first introduced
by Michie & Bodenheimer (1963) and intensively discussed in King (1962) and King (1966)
who use them to fit old globular clusters like it was done by Plummer (1911). The idea
C Stellar number density profiles
is based on the model of the isothermal spheres but considers a finite tidal radius of the
cluster rtide , beyond which the cluster ceases. It is called tidal radius because for distances
r > rtide from the cluster center, the tidal forces of the hosting galaxy are able to remove
stars from the cluster. The energy distribution function of the King model in dependence of
the relative energy ε = Φ−1/2v 2 and relative potential energy of the particles Ψ = −Φ+Φ0
is given by

2
ρ (2πσ 2 )−3/2 (eε/σK
− 1)
1
K
fKing (ε) =
0
ε > 0;
(C.3)
ε≤0
which represents the distribution function of an isothermal sphere but less dense at large
cluster radii, where ε is small. The distribution function can be integrated over the stellar
velocities to obtain the density in dependence of the relative King cluster potential Ψ(r)
"
ρKing (Ψ(r)) = ρ1 e
2
Ψ(r)/σK
erf
! s
p
#
Ψ(r)
2Ψ(r)
4Ψ(r)
1+
−
,
2
2
σK
πσK
3σK
(C.4)
where σK is the King velocity dispersion and erf the error function. For setting up a
cluster with a King density distribution ρKing one needs to numerically integrate the Poisson
equation for Ψ(r) (see Binney & Tremaine, 2008, for a complete introduction into King
models).
c=
c=
W0
Figure C.1: King concentration parameter c and King radius r0 relative to half-mass radius as a function of the
dimensionless King Parameter W0 adapted from Vesperini (1994).
Furthermore, the King models can be parametrised by their concentration c = log10 (rtide /r0 )
or the ratio of
124
W0 =
with the King radius of the cluster given by
s
r0 =
Ψ(0)
,
σK
(C.5)
2
9σK
,
4πGρ0
(C.6)
which defines the size of the core of the model. The relation between the King concentration
c and the King parameter W0 is plotted in Figure C.1.
The advantage of the King models hereby is that the King parameter W0 can now be
chosen according to the desired cluster type. For a high King parameter W0 (or concentration c) a rather flat distribution in the core region and a quick drop in density at the outer
cluster region is obtained, making such King models an ideal candidate to fit observed
surface brightness profiles of old globular clusters (King, 1962, 1966). Moreover, Plummer
models can even be approximated by a King model with King parameter W0 = 4. In the
limit W0 → ∞ (c → ∞) the King models converge to the isothermal sphere model.
125
C Stellar number density profiles
126
D Gas expulsion models
Varying background potential
One further approach to simulate a surrounding gas cloud is the implementation of a background potential, which increases the total potential energy of the cluster. In nbody6 this
has been done by implementing a constant background Plummer profile, which depends
on the initial gas mass mgas and the Plummer radius a (a detailed description of Plummer
density profiles is given in Appendix C). However, after a reasonable time span the gas is
expelled from the cluster, thus, the background potential has to decrease after a predefined
time delay tdelay . Furthermore, the expulsion is not inevitably instantaneous but might be
elongated over several crossing times. Geyer & Burkert (2001) assumed a linear decay of
the external gas potential by a time-dependent multiplier



1


ξ = 1 − c · (t − tdelay )/texp



0
t < tdelay
tdelay < t < tdelay + texp ,
(D.1)
t > tdelay + texp
where texp is the time scale for the expulsion process and c = 1 in their study.
There are two intuitive methods to truncate the gas profile, i.e. to simulate the expulsion
of the gas: (i) a temporally varying gas mass mgas (t), which is consistent with the approach
of Geyer & Burkert (2001), as well as (ii) a spatially evolving gas model a(t). So far there
are no science-based restrictions for the profile of the gas loss. One such approach to define
a time-dependent gas mass loss is given by (Eq. D.1)
mgas (t) = mgas (0) · ξ(t, c)
with c = 1,
(D.2)
with c = −1,
(D.3)
and a linearly increasing Plummer radius
a(t) = a(0) · ξ(t, c)
while e.g. Kroupa et al. (2001) assumed an exponential decay of the gas mass and c = −1/2
for a time-varying Plummer radius.
D Gas expulsion models
Radially-varying star formation efficiency
For simplicity reasons numerical studies use a global star formation efficiency for embedded
cluster models. In fact, the star formation efficiency is rather a function of the radius
ǫSFE := ǫ(r), with a higher star formation efficiency in the central cluster regions.
Recently, Parmentier & Pfalzner (2012) analysed the evolution of the local star formation
efficiency ǫSFE in young clusters as a function of the cluster age and the local gas density
(radius). Assuming the cluster as a symmetric sphere, its initial volume density ρ0 is given
by
ρ0 (r) =
3 − p0 Mtot −p0
3−p0 r
4π Rtot
(D.4)
with total cluster mass Mtot (= Mgas initially), cluster radius Rtot , and density distribution
index p0 ∈ [1.4, 2.2] (Beuther et al., 2002; Mueller et al., 2002; Pirogov, 2009).
The spherical molecular cloud can be discretised into infinitesimal mass shells. The mass
of these shells can be described by
dmgas (ti , r) = dmgas (ti−1 , r) − ǫf f ·
ti − ti−1
· dmgas (ti−1 , r)
τf f (ti−1 , r)
(D.5)
dmstar (ti , r) = dmstar (ti−1 , r) + ǫf f ·
ti − ti−1
· dmgas (ti−1 , r),
τf f (ti−1 , r)
(D.6)
where the initial shell mass is assumed to be dm0 (0, r) = dmgas (t, r) + dmstar (t, r) for any
time step t and shell radius r. dmgas (t, r) is the gas mass of an infinitesimal shell and
dmstar (t, r) the stellar mass fraction, respectively. The star formation efficiency per freefall time ǫf f , which gives the fraction of gas mass that is turned into stars per local free-fall
time τf f (t, r), is defined constant while the local free-fall time is density-dependent, given
by
τf f (t, r) =
s
3π
.
32Gρgas (t, r)
(D.7)
The star formation in the center of the cloud proceeds much faster than in the outer parts
due to the much denser natal gas. Here, this yields a steeper density profile of the stars
than in the case of the initial molecular cloud. The dependence of ρgas on the evolved
time is due to the depletion of the surrounding gas resulting in a deceleration of the star
formation process.
Equations D.5 and D.6 are first order differential equations which translate to
ǫf f
∂ρgas (t, r)
=−
ρgas (t, r)
∂t
τf f (t, r)
128
(D.8)
ǫf f
∂ρstar (t, r)
=
ρgas (t, r).
∂t
τf f (t, r)
(D.9)
By using Eq. D.7 and integrating one obtains
ρgas (t, r) =
ρ0 (r)
ρstar (t, r) = ρ0 (r) −
−1/2
+
r
8G
· ǫf f · t
3π
ρ0 (r)−1/2 +
r
!−2
8G
· ǫf f · t
3π
(D.10)
!−2
.
(D.11)
Finally, this results in a local star formation efficiency of
ǫSFE (r) =
ρstar (r)
=1−
ρstar (r) + ρgas (r)
ρ0
(r)−1/2
+
q
8G
3π
ρ0 (r)
· ǫf f · t
−2
.
(D.12)
Considering Eq. D.4 the local star formation efficiency in Eq. D.12 now depends only
on the initial (total) mass Mtot , the cluster radius Rtot , the initial gas density distribution
index p0 , the star formation efficiency per free-fall time ǫf f , and the time of evolution t.
Similar to the instantaneous gas expulsion process the virial ratio, given by Eq. 2.32, can
be used to implement the radial dependence of the star formation efficiency in nbody6 by
Q(r) =
Q0
.
ǫ(r)
(D.13)
Due to a subsequently implemented scaling routine the modified velocity profile can in
addition be scaled to arbitrary global star formation efficiency ǫ.
129
D Gas expulsion models
130
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144
Erklärung
“Ich versichere, dass ich die von mir vorgelegte Dissertation selbständig angefertigt, die
benutzten Quellen und Hilfsmittel vollständig angegeben und die Stellen der Arbeit – einschließlich Tabellen, Karten und Abbildungen –, die anderen Werken im Wortlaut oder
dem Sinn nach entnommen sind, in jedem Einzelfall als Entlehnung kenntlich gemacht
habe; dass diese Dissertation noch keiner anderen Fakultät oder Universität zur Prüfung
vorgelegen hat; dass sie – abgesehen von unten angegebenen Teilpublikationen – noch nicht
veröffentlicht worden ist sowie, dass ich eine solche Veröffentlichung vor Abschluss des
Promotionsverfahrens nicht vornehmen werde. Die Bestimmungen der Promotionsordnung
sind mir bekannt. Die von mir vorgelegte Dissertation ist von Frau Prof. Dr. Susanne
Pfalzner betreut worden.”
Bonn, den 25. März 2013
(Manuel Steinhausen)
Acknowledgements
First and foremost, I would like to express my special thanks and deepest gratitude to
Prof. Dr. Susanne Pfalzner for giving me the opportunity to write my PhD dissertation
under her supervision. Her constant support and valuable guidance have been a remarkable source of motivation during all this time. I would like to express my gratitude for her
invaluable scientific insight, comprehensive knowledge and penetrative approach, that have
on all counts shaped my dissertation.
Furthermore, I am sincerely grateful to Prof. Dr. Andreas Eckart, head of the I. Physikalisches Institut in Cologne, and Prof. Dr. Karl Menten, head of the Max-Planck Institut für
Radioastronomie in Bonn, for their hospitality and their financial support. In particular,
I would like to thank the Max-Planck Foundation for awarding me with a stipend of the
Max-Planck-Gesellschaft.
I am indebted to Christina Korntreff, Andreas Breslau, Kirsten Vincke, Anastasia Tsitali and Guang-Xing Li for many enlightening conversations and helpful comments on my
work. It was a pleasure to work and socialize with you on a daily basis. Special thanks go to
Dr. Thomas Kaczmarek and Dr. Christoph Olczak for their expert advice and instructive
suggestions since the start of my studies until today.
I would like to extend my gratitude to the entire Eckart workgroup and Menten workgroup
for the friendly atmosphere and for offering me an insightful glance into the exciting work
in their respective fields.
Last but not least, I would like to thank my family, my girlfriend and everyone who supported me in one way or another in the different stages of my research.
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